Brief User Guide for TI-84 Statistics

Contents: This document covers single- and two-variable statistics, scatter plot, regression analysis, normalpdf and
 normalcdf, Student's Distribution, geometric mean, and much more.
 Date Last Revised: 5/9/15

INDEX:
For TI-83 Plus  statistics click here.

To facilitate lookup, the instructions are divided into the different categories. To find a category or function
that you're interested in, set the cursor after the index and use the browser Find function.
NOTE: This page is suitable for viewing on mobile devices.

PART I: BASIC OPERATIONS
 I.   Data Manipulation - Sorting data, clearing lists, Entering data, ... 

NOTE: In the following material, Part II is for obtaining only an answer; Part III is for those who
         must provide intermediate calculations.     

   PART II: USING THE CALCULATOR FOR A COMPLETE SOLUTION:
 II.  Single-Variable Statistics -  simple histogram with the calculator,  sorting data into classes, 
             choosing your own classes when using the calculator, frequency polygon, cumulative frequency (Ogive), percentile
             graph, relative frequency polygon, cumulative relative frequency graph, histogram from grouped data, 
             frequency and cumulative frequency graphs from grouped data, box and whisker plot, discrete probability 
            distribution, coefficient of  variation, finding standard deviation, finding standard deviation from grouped 
            data, standard deviation with a computation formula, weighted average, median of grouped data, geometric mean.
 III.  Probability: Factorials, random numbers, permutations, combinations, geometpfd, geometcfd, hypergeometric
        probability 
 IV.  Two-Variable Statistics – scatter plot, regression analysis, finding r, r², a, and b in correlation using a calculator, testing
              the correlation coefficient,
  V.  Normal Distribution - Area under a normal curve, Finding Z values, Graphing a curve,  WINDOW 
        settings for graphing a curve, Probability Distribution Function using normalpdf(, Graphing the 
        Normal Distribution Using normalpdf(, normalcdf(,  and ZInterval,
VI. Other Distributions -  Finding a TInterval, Student's t Distribution, Using invT to Find a t-value given α 
       and df, Chi-squared Distribution, binomialpdf, binomialcdf.
VII. Hypothesis testing - mean and z-test (data), mean and z-test (statistics), mean and t-test (data), 
              mean and t-test (statistics).
VIII. Statistics of two Populations - confidence interval for two dependent population, confidence interval for two 
              independent populations (Data and Stats),  standard error of estimate
   IX.  Other Tests and Inferences -  one-way ANOVA, Chi-Square test for independence, X² Goodness of Fit,

Part III: DOING PROCEDURES BY HAND WITH THE AID OF A CALCULATOR   
          NOTE: These procedures are similar to those above, but they are for those who are required to show their
            work rather than just an answer.
 X.  Aids in doing statistics by hand - Doing list arithmetic,  Finding Ʃx² , (Ʃx)^{2 , etc using lists,}
XI.  Single-Variable Statistics - Histogram by hand
  Doing a Discrete Probability Distribution by Hand
XII. Two Variable Statistics -  finding r, r², a, and b in correlation using a computation formula,
 Finding the Values a and b for the Best-Fit Equation Using a Computation Formula:

      APPENDIX:  Program for sorting data into classes.
 

RELEASE DATE:  9/25/15        DATE LAST REVISED:  10/5/15

 © 2015 Frank Kizer     NOTE:  Copying restrictions and printing hints are at the end of this document.

FORWARD:  It seems that at the ends of the spectrum of opinions about using calculators there are two polar 
opposites:  Use a calculator to the maximum or don't use it for anything except arithmetic.  I have tried to take
into consideration the broad spectrum and include methods that use only a calculator and those that use the 
calculator to take some of the drudgery of arithmetic out of the use of the computation formulas. The methods
that belong in the different approaches are separated into Part II and Part III.

PART I: BASIC OPERATIONS

I.  Data Manipulation
    SETUP: To set up the calculator, press MODE and highlight MATHPRINT; then turn on STAT DIAGNOSTICS and
          STAT WIZARD.
      1) Clearing Lists:
          In some instances you may want to clear a list or lists before you start entering data.  You   
          can overwrite data already in a list, but remember that if the old list was longer than the new one, 
          you must delete the remaining old data an item at a time.  The easiest way to clear one of the tabular
          lists, L ₁ -L ₆ is to place the cursor on the name above the list and press CLEAR; then ENTER. DO NOT press Del.
         That will delete that list from the display. You can also clear a number of lists or any list as follows: a) Press
         STAT, 4 (ClrList).  This will paste "ClrList" to the home screen.  Press 2nd; then the button for the list number you
         want to clear, for example L₁ ; then press ENTER.  If you want to clear more than one list, separate the lists by
         commas. 
      2)  Entering Data:
          a)  Press STAT; then ENTER.  Tables for entering data will appear.
          b)  To enter data, just place the cursor where you want to enter the data and press the correct 
               numbers.  You don’t have to erase old data if there is already data in the list, but if the old list 
               is longer than the new list, you will need to delete the remaining old data items.  Just place 
               the cursor over the data and press DEL.       

    3)  Putting Data in Order:
         In some of the procedures below, you may need to put the data in order of value.  You can do 
         do that as follows:
           a)  Press STAT, 2 (SortA).  This will paste SortA to the home screen.  
           b)  Press 2^nd, L₁ (or whatever list you want to sort); then press ENTER. Finally, return to your 
                tables to view the sorted data. Note that you can also sort data in descending order with 
                SortD.

PART II: USING A CALCULATOR FOR A COMPLETE SOLUTION:

   II. Single-Variable Statistics:
      1) Graphing or Drawing a Frequency Histogram (Ungrouped Data):
           I have included two methods for graphing a histogram The first 
           method allows the calculator to calculate the class limits and boundaries.  The second method requires some
           involvement in the procedure by the calculator user, although the calculator does all of the arithmetic.
           Finally, I have included a calculator program in the APPENDIX to sort the data into classes.
          A)  Entering Data:
              a) Press STAT; then ENTER.  Tables for entering data will appear.  You may want to completely clear the list 
                  you are planning to use by moving the cursor to the title, for example L₁, and pressing CLEAR, ENTER. 
                   DO NOT press DEL while you have the title highlighted or you will delete that list from the tables.  
              b)  To enter data, just place the cursor where you want to enter the data and press the correct 
                   numbers. If you have not cleared the list and the old list is longer than the new list, you will need to 
                   delete the remaining old data items.  Just place the cursor over the data you want to delete and press DEL. 
        
         B) Doing the Histogram by Letting the Calculator Choose the Class Limits:
                Preparing the Calculator
                   a)  Go to the lists and enter data by pressing [STAT], [ENTER] to display the list tables. 
                   b)  Enter the numbers in L₁, or whatever list you choose, using the method under Entering Data immediately
                        above.
                  c)  Press [2nd], [STAT PLOT] and press [ENTER]. Press [ENTER] again to turn on Plot 
                       1 .
                  d)  Move the cursor to the icons opposite Type, select the third icon, histogram, and press [ENTER] to  
                       highlight the histogram icon.
                  e)  Enter L₁ (or whatever list your data is in) opposite Xlist, by pressing 2nd, L_1.  Make sure there's a
                      1 opposite Freq if you have ungrouped data. You may need to press ALPHA to toggle the mode to numbers
                      to enter a 1.
                  f)  Press [ZOOM]; then 9 (ZoomStat) and the histogram will appear on the screen.
                  g)  If you want to know the class limits and the number of data points in each class, press TRACE and 
                       move the cursor across the tops of the bars.
                  h) If you want the class boundaries rather than the class limits, Press WINDOW, enter
                       -.5 (minus .5) after the value for X-min and press ENTER. A new value for X-min will be 
                       displayed. Press Graph to plot the histogram with the boundaries.
         

         C)  Doing the Histogram by Choosing Your Own Class Widths:
               Suppose that you have the numbers listed below and you want to construct a histogram with five classes.
               Numbers: 22, 23, 25, 27, 20, 33, 37, 38, 40, 34, 28, 29, 31, 42, 43, 40

             a) Press Y= and deselect or clear all entries in the Y= positions.
             b) Press 2^ND, STAT PLOT (the Y= key) to go to the plot functions. Press ENTER twice to turn on plot 1.
             c) Scroll to the third icon opposite "Type" and press ENTER to highlight that icon.
             d) Opposite "Xlist," if L₁ is not already entered, enter it by pressing 2ND, L₁.  Opposite "Freq" enter 1 if it's not
                 already there. If an “A” is   blinking in the cursor, press ALPHA.
             e) Press ZOOM, scroll down to ZoomStat and press ENTER.  This will display a preliminary histogram with
                 the number of classes determined by the calculator.

               To adjust the histogram with the number of classes you want and a more acceptable display, do the following:
              f) Press WINDOW and move the cursor opposite Xmin. Press 2^ND, LIST, move the cursor to MATH, and press
                  ENTER to display min(.  Press 2^ND, L₁ (the 1 key) to display min(L₁ opposite Xmin.  Press ENTER to display
                  Xmin which in our case is 20.
              g)Repeat the procedure for Xmax except when you go to MATH, select 2:max(. The number for Xmax will be
                  43 for our problem.
              h) Oppose Xscl, enter the expression (43-20)/5. The answer will be 4.6 when you move the cursor from Xscl.
                  Round the number up to 5 and replace 4.5 with 5.
               i) Change the number opposite Ymin to -1 and the number opposite Ymax to 5. If you wish, you can  scroll
                  down to ∆X and change the number to .3. That will give you a little space to the right of the graph. Press
                  Graph to display the new histogram.
              j) To get the min and max of a class, and the number of data points in a class, press TRACE and move the cursor
                  across the tops of the bars. Note that the max for the class will be displayed as < some number. Use the next
                  lower number for the class max. In this case that's 24 for the first class. Move the cursor and record the max
                  and min and the value for "n" for each class.
            k) To find the midpoint use the formula (Class max + Class min)/2.
                In this instance that will be (24+20)/2 or 22. Now add successive values to that midpoint to get the other
                midpoints. For example, the midpoint for the second class will be 22+5=27 and for the third class will be
                27+5 = 32.
            l) If you want to find the class boundaries, go to WINDOW and change Xmin to 19.5. Use TRACE again, but
                now use the min and max numbers displayed as the boundaries.

        E)  Using my program NOSTOCLS to sort the data into classes:
                    There are times when it would be useful to check data for two or three different classes.  This might be useful
                    for grading papers or for saving class time for something more important than tallying.  With my program,
                    this can be done in about one minute after entering the data in  list L_1. Suppose we have a certain set of  60
                    numbers with values from 1 to 47. We want to check the distribution for six, seven, and eight classes.  The
                    program will quickly give the distributions, 9,14, 17, 6, 5, 3, 3, 3 for a class width of 6; 11,18, 14, 7, 3, 4, 3 for
                    class width of 7; and 14,21,11,6,4,4 for class width of eight.  The program is included in the APPENDIX at the
                    end of this document.  

    2.  Constructing a Frequency Polygon from Ungrouped Data:
          After graphing the histogram, you can use TRACE to get the data for the frequency polygon and a cumulative
          frequency graph if you wish.  
          a)  Press TRACE and use the arrow to move across the histogram bars.  Record the values for x-min, x-max, and "n"
                on a sheet of paper in tabular form.
          b)  Add one-half the class width to each x-min value and record those values.   Store these values in a list, for
                example L₂ if you have your histogram data in L₁.  Store the corresponding values of "n" in L₃.
          c)  Press 2nd, STAT PLOT, ENTER.  If "On" is not highlighted; then select it and press ENTER.
          d)  Highlight the second icon on the first row; then enter L₂ opposite Xlist and L₃ opposite Ylist.
          e)  Press ZOOM, 9 and the graph will appear on the screen.
          NOTE:  Some teachers or texts prefer return-to-zero graphs.  If your course requires that, do the following after
              step b) above:
          A. Calculate a midpoint of a new class preceding the first class and another midpoint after the last class.  These 
               values will be entered into L₂.  To do that place the cursor at the first item in L₂, press INS and replace the zero that  
               appears with the first midpoint you calculated. Go to the bottom of the L₂ list and enter the second value you 
               calculated. 
         B.  Now you want to enter zero in L₃ opposite each of these new midpoints.  Place the cursor at the top of L₃ and press 
              INS.  A zero will be added.  Now cursor to the bottom of the list and enter a zero opposite the last new midpoint
              that you entered in L₂. 
         C. Proceed with step c) above. 

     3.  Constructing a Cumulative Frequency Chart (Ogive) Graph:
          a)  Enter the Xmax values that you recorded above in a list.  For example, L₄  if you still have data in the  other  lists. 
          b)  Now, store the cumulative frequency data in L ₅ as follows:  Place the cursor over the list title, L₁.  Press 2nd, LIST, 
                cursor to OPS, and press 6.  The expression cumSum( will be displayed at the bottom of the screen.  
          c)  With the cursor after the parenthesis, press 2nd, L_3, ), _, ENTER.  You will now have cumSum(L₃)
               at the bottom of  the lists screen. 
          d)  Press 2nd, STAT PLOT, highlight "On" if necessary and press ENTER 
          e)  Highlight the second icon on the first row; then enter L₄ opposite Xlist and L₅ opposite Ylist.
               NOTE:  If you did a return-to-zero graph for the frequency polygon, go to the list and delete the last
               midpoint and zero in L₄ and L₅ respectively.
          f)  Press ZOOM, 9 and the graph will appear on the screen.

     4)   Relative Frequency polygon and Cumulative Relative Frequency (Ogive) Graphs:  
            These are done similarly to the frequency polygon.  After storing the data for the x-values and frequencies, do the
            steps listed for each type graph.
            Relative Frequency:
            Assume that we want to store the relative frequencies in list L₃, the frequencies are in L_2,and the x-values  are in L₁ .
            a)  First place the cursor to highlight the list title, L₃.  Press 2ND, L₂, ÷, 2nd, LIST, move the cursor to MATH and press 5.  You 
                  should now have L₂/sum( displayed on the bottom of the list screen.
            b)  Press 2ND, L₂, ), ENTER and the relative frequencies will be stored in list L₃.
             c)  To plot a graph of the relative frequency, press 2nd, STAT PLOT, ENTER.   If "On" is not highlighted, select it and
                  press ENTER.
            d)  Highlight the second icon, and enter L₁ opposite Xlist and L₃ opposite Ylist.
             e)  Press ZOOM, 9 and the graph will be displayed. 
            Cumulative Relative Frequency:
            Assume that we want to store the cumulative relative frequencies in list L₄ and that the relative frequency
             is still stored in L₃ from the above relative frequency operation above, and that the x-values are in L₁.
            a)  First place the cursor to highlight the list title, L₄.  Press 2ND, LIST, move the cursor to OPS and press 6. 
                 You should now have cumsum( at the bottom of the screen.   
            b)  Press 2ND, L₃, ), ENTER. The cumulative relative frequencies will now be stored in L₄.
             c)  Press 2nd, STAT PLOT, ENTER.   If "On" is not highlighted, select it and press ENTER.
            d)  Highlight the second icon, and enter L₁ opposite Xlist and L₄ opposite Ylist.
             e)  Press ZOOM, 9 and the graph will be displayed.

      5)  Histogram Using Grouped Data:
           a)  Enter the midpoints of the classes into L₁ and the corresponding frequencies into L₂ . You
                can enter the arithmetic operations in the list if you prefer, e.g. (upper-lower)/2
           b)  Press 2nd, STAT PLOT, ENTER.
           c)  If "On" is not highlighted, select it and press ENTER.
           d)  Move the cursor to the histogram symbol and press ENTER; then enter L₁ opposite Xlist and L₂ opposite Freq.
           e)  Press ZOOM, 9 and the histogram will be displayed.  
             Note:  If you want to select your own classes do the following before pressing ZOOM 9 in step "e" above.
              1)  Press WINDOW and enter the lowest boundary value opposite Xmin
                    and  the class width opposite Xscl.  You may also want to change Ymin to something like zero or -1
                    so that histogram will not be so far above the baseline.  Further, you may want to set Xmax to a value
                   slightly above the last class boundary.
              2)  Press GRAPH and the histogram will be displayed.

     6)  Frequency Polygon Using Grouped Data:
           Do this exactly like the histogram, except select the line graph icon, the second icon.  If you've already done the    
           histogram, just change the icon and press GRAPH.

     7)  Cumulative Frequency (Ogive) Graph from Grouped Date:
           a)  Enter the  class upper boundaries in a list, for example, L₃ if you have data in the first two lists.
           b)  If you have the frequency in L₂ , place the cursor over the list title, L₄, and  do the following:  
                A)   Press 2nd, LIST, cursor to OPS, and press 6.  cumSum(  will be displayed at the bottom of the list screen.  
                B)  With the cursor after the parenthesis, press 2nd, L_2, ) .  You will now have
                      cumSum(L₂) at the bottom of the lists screen.  Press ENTER.
          c)  Press 2nd, STAT PLOT, ENTER.   If "On" is not highlighted, select it and press ENTER.
          d)  Highlight the second icon, and enter L₃ opposite Xlist and L₄ opposite Ylist.
          e)  Press ZOOM, 9 and the graph will be displayed.

     8)  Relative Frequency and Cumulative Relative Frequency Graphs for Grouped Data:
            These are done similarly to the as frequency polygon.  After storing the data for the midpoints and frequencies,
            do the steps listed for each type graph.
            Relative Frequency:
            Assume that we want to store the relative frequencies in list L₅ and the upper limits on the classes are in L₃ .
            a)  First place the cursor to highlight the list title, L₅.  Press 2ND, L₃, ÷, 2nd, LIST, move the cursor to MATH and
                 press 5.  You should now have L₃/sum( displayed on the bottom of the list screen.
            b)  Press 2ND, L₃, ), ENTER. 
             c)  Press 2nd, STAT PLOT, ENTER.   If "On" is not highlighted, select it and press ENTER.
            d)  Highlight the second icon, and enter L₃ opposite Xlist and L₅ opposite Ylist.
             e)  Press ZOOM, 9 and the graph will be displayed. 
            Cumulative Relative Frequency:
            Assume that we want to store the cumulative relative frequencies in list L₆ and that the relative frequency
             is still stored in L₅ from the above relative frequency operation above, and that the class upper boundaries are in L₃.
            a)  First place the cursor to highlight the list title, L₆.  Press 2ND, LIST, move the cursor to OPS and press 6.  You should now
                  have cumsum( at the bottom of the screen.   
            b)  Press 2ND, L₅, ), ENTER. The cumulative relative frequencies will now be stored in L₆.
             c)  Press 2nd, STAT PLOT, ENTER.   If "On" is not highlighted, select it and press ENTER.
            d)  Highlight the second icon, and enter L₃ opposite Xlist and L₆ opposite Ylist.
             e)  Press ZOOM, 9 and the graph will be displayed.

      9)  Percentile Graphs:
            This graph is fairly similar to the Ogive graph.  We will do this in two groups of steps:  Preparing data
            and plotting data.
            Preparing Data:
            a)  Enter upper boundaries in L₁ and the corresponding frequencies in L2.  If you want the graph to start 
                  at zero, enter the first lower boundary with zero for the frequency.
           b)  Highlight L₃ at the top of the lists on the LIST screen.
           c)  Press 2nd, LIST, move the cursor to OPS, and press 6 to paste cumSum( to the bottom of the list screen.
           d)  Press 2nd, L₂ , ), ÷ .  You now should have cumSum(L₂)/ at the bottom of the list screen.
           e)  Press 2nd, LIST, cursor to MATH and press 5 to paste sum( to the list screen.
           f)  Press 2nd, L₂, ).  You now should have cumSum(L₂)/Sum(L₂) at the bottom of the list screen.
           g)  Press x (the multiply symbol), 100.  You now should have cumSum(L₂)/Sum(L₂) *100 at the bottom 
                 of the list screen.  
           h)  Press ENTER and the data will be stored in L₃ .
          Plotting the Data:
           i)  Press 2nd, STAT PLOT, ENTER
           j)  Select the second icon and enter L₁ opposite Xlist and L₃ opposite Ylist.
           k)  Press ZOOM, 9 and your graph will be displayed.
           l)  You can find the exact percentiles of the boundaries by using TRACE, and approximate percentiles of 
                other x-values by using the cursor.
         
    10) Box and Whisker Plot
            NB:  Users should be aware that there is no standard way of finding quartiles.  The TI uses the Moore and 
                     McCabe (M-and-M) method.  Minitab and Excel use different methods. If you get different answers
                     from those in your book, check to see if your book is using a method other than M-and-M. 
         a)  First go to the graphing screen by pressing the Y= button.  Deselect any  Y= functions so that  
               they won't be entered on your graph.  If you choose, clear the list as described at the beginning 
              of this document.
         b)  Press [STAT], [ENTER] to go to the list tables. 
         c)  Enter your  numbers in L1.  (Or whatever list you choose.)
         d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn on  Plot 1.
         e)  Opposite the word Type, cursor  to the icon that represents a box-and-whisker plot, icon 5, and  
              press [ENTER] to highlight the box plot icon.  (See the note at the end of this topic for when to 
              use icon 4.)
         f)  Enter the list you put the data in, usually L₁, in the Xlist, by pressing 2nd, L_1. or whatever list
             you chose.  Make sure the number 1 is opposite Freq.
         g)  Press [ZOOM]; then 9 (ZoomStat)  and the box-and-whisker plot will appear on the screen.
         h)  To find the numbers for the limits of the quartiles, press [TRACE]; then use  the cursor to move 
               across the diagram and obtain the values for quartiles or the beginning and ending values. 
              NOTE:  If you have one or two outliers (numbers much larger than the rest) you may want to use   
             icon 4.  This will not include the outliers in the last whisker, but will plot them as separate points 
             after the end of the last whisker.

    11)  Calculation of Coefficient of Variation from List Data:
           The coefficient of variation, CV=s/x-bar, is a simple arithmetic calculation if you have the mean 
           and standard deviation.  But calculations from a list are a little more involved.  Here's an easy way
            to do it.
            a)  Store the data in a list, for example L₁, and move the cursor to the first blank space at the end of the data. 
            b)  Press 2nd, LIST and move the cursor to MATH.
            c)  Press 7 to paste StdDev( to the bottom of the list  screen.
            d) Press 2nd, L₁, ), and then press the divide symbol.
             e)  Press 2nd, LIST, move the cursor to MATH, and press 3 for mean.
              f)  Press 2nd, L₁ , close the parentheses and then.  You should now have StdDev(L₁₎/mean(L₁).  Press ENTER to display the 
                   CV as the last number in L₁.  If you want CV in percent, multiply this number by 100. 
              NOTE: If you're going to use this list for other calculations, be sure to delete the CV value before performing any operations.

   12)  Finding the Standard Deviation and Variance of Ungrouped Data:
          a) Entering Data:
            1)  Press STAT; then ENTER.  Tables for entering data will appear.  If you need to clear a     
                 list, move the cursor up to highlight the list name; then press CLEAR, ENTER.
            2)  To enter data, just place the cursor where you want to enter the data and press the  
               correct numbers, then press ENTER.  You don't have to erase old data if there is  
               already data in the list, but if the old list is longer than the new list, you will need to 
               delete the remaining old data items.  Just place the cursor over the data and press 
               DEL. 
        b) Suppose that you have the sample of data listed immediately below and you want to find   
             the standard deviation and variance.
             Data:  22, 27, 15, 35, 30, 52, 35
       c) Enter the data in list L₁ as described under Entering Data immediately above.
       d) Press STAT, move the cursor to CALC, and press ENTER. The expression “1-Var Stats” 
            should be pasted to the home screen along with the following list:
             List:
             FreqList:
             Calculate
            Enter the list, for example L₁, where the data is stored opposite list by pressing 2^nd and the key number for the list where the data is stored.
          e) It you have entered repeats with a frequency, move the cursor opposite FreqList and enter the frequencies opposite the appropriate numbers in an adjacent list. For example, if the data is in L₁; then enter the frequencies in L₂.
          f) Move the cursor to highlight "Calculate" and press ENTER.  The standard
            deviation and several other statistics will be displayed.
         e)  To calculate  the variance, merely re-enter the value for the  standard deviation, S_x  or σ_x depending on whether you are using the population or sample , 
              and square it^,   Note:  If you round off the standard deviation, you may have a slightly different answer than 
              you would if you had calculated the variance separately by hand.  To avoid that, enter all decimal places for S_x and 
             square that value.  If you don't like entering long numbers, you can do this:  Press VARS, 5, 3, ENTER, x² , ENTER.          

13.  Finding the Variance and Standard Deviation of Grouped data.
    A.  Calculated by the Calculator Only:
      a)  Entering Data:
          1)  Press STAT; then ENTER.  Tables for entering data will appear.  If you need to clear a     
               list, move the cursor up to highlight the list name; then press CLEAR, ENTER.
          2)  To enter data, just place the cursor where you want to enter the data and press the  
               correct numbers and press ENTER.  You don't have to erase old data if there is already  
               data in the list, but if the old list is longer than the new list, you will need to delete the  
               remaining old data terms.  Just place the cursor over the data and press DEL. 
      b)  Suppose that you have the sample of data listed in the table below and you want to find 
            the standard deviation and variance.
        

      c)  Enter the class midpoints in list L₁.  You can either do the midpoints by hand or calculate   
            and store them in list L₁ as follows:
            (1) Store the lower boundaries in list L₁ and the upper boundaries in L₂.  
            (2) Place the cursor on the title of L₁; then press (, 2ND, L₁, + 2ND, L₂,), ÷, 
                  2.  You should have (L₁ + L₂)/2 at the bottom left of the tables.  Press     
                  ENTER and the midpoints will be stored in L₁.
      d)  Clear list L₂ if necessary, and enter the frequencies in L₂ as described under Entering Data immediately above.
            Now we’ll calculate the required statistics.
      e) Press STAT, move the cursor to CALC, and press ENTER. The expression “1-Var   
          Stats” should be pasted to the home screen along with the following list:
               1-Var Stats
             List:
             FreqList:
             Calculate
          You must enter the list where the data is stored opposite List. Press 2^nd and the key 
          number for the list where the data is stored. Do the same to enter the frequencies
          in list L2 opposite FreqList.
        f) Move the cursor to highlight "Calculate" and press ENTER.  The standard
           deviation and several other statistics will be displayed. The sample standard deviation is  
           14.868….
        g)  To find the variance, just square the standard deviation by entering the number, pressing 
            the x² button, and then ENTER. 

14)  Weighted Average:  
          Suppose you have some scores with the weights indicated:  
          Score      Weight
           83              .3
           85              .3
           85              .5
           89              .3
           90              .7
           a)  Press STAT, ENTER and enter the scores in list L₁ and the weights in L₂.
           b) Press STAT, move the cursor to CALC, and press ENTER to paste "1-Var Stats"  to the home screen.  
           c)  If your scores and weights are in lists as indicated above, press ENTER and the weighted average will be 
                given as x-bar (x with a bar over it.).  If your data are in other lists, enter those lists separated by a comma and 
                press ENTER.

15)  Median of Grouped data:
       Consider the following table.

a)  To find the median class, enter the midpoints in L₁ and the frequencies in L₂ .
b)  Press STAT, move the cursor to highlight CALC and press ENTER. 
     The following will be displayed on the home screen. 
        1-Var Stats
     List:L₁
     FreqList:L₂
     Calculate
c  If the list numbers are in other lists, you must enter the correct lists.  For example,   
     press 2^ND, L₂, comma, 2^ND, L₃ and then press ENTER
d)  Move the cursor to Calculate, press ENTER and scroll down to Med=19.5.  That is the median of the class 15-  
     24. So 15-24 is the median class. 
e)  Enter the appropriate data into the following formula: 
Median = L + I *(N/2 - F)/f
Where
L = lower boundary of the interval containing the median. 
I = width of the interval containing the median. 
N = total number of respondents. 
F = cumulative frequency of those below the median class. 
f = number of cases in the median class.
f)  When you are finished entering, you should have this:
    14.5+10(5000/2-750)/2005
g)  Press ENTER and you should get 23.228… Notice that the answer is different form 
     the value of 19.5 given by the calculator. That value of 19.5 was chosen by merely finding the  midpoint  
     of the median class.

16) 3. Geometric Mean:
     Let's do the geometric mean of the S&P 500 as listed in the table at the beginning below.
    

COMMENT: Can we just calculate the arithmetic mean?  No, this is not a number of different investments calculated over the same period of time, so that we could just add them and divide
by the number of investments. It's one investment over different one-year periods. So, we must use the geometric mean.
     a) Press STAT, ENTER and enter the returns data in list L₁ or whatever list is convenient.
     b) Place the cursor on the list name of list L₂ or whatever list you choose, and enter 2nd, L₁/100 +1 and press ENTER.
         Now, we're going to use the formula (products of L₂)^1/k , where "k" is the number of return values. In out case that's
          10.
     c) Press 2nd, STAT, move the cursor to MATH on the menu that appears and press 6 for prod(.
     d) Press 2nd, L₂ so that you have prod(L₂ on the display. Press ENTER.
      e) Now enter the following: (Ans)^1/10 -1 . You may need to enter the entire exponent in parentheses if you have
           an older calculator. If you did the S&P returns, you should get 7.4.

III.  Probability: Permutations, combinations, factorials, random numbers:
 

 1.  Randomly generated data sets:
      Sometimes problems use a randomly generated set of data. Suppose we want to generate 10  
      random numbers between 1 and 50 and store them in List 1.  The proper syntax is randint(lower, 
      upper, how many).  That can be obtained as follows:
      a)  Press MATH, cursor over to PRB and press the number 5. randint( will appear on the screen.
      b)  Enter 1, 50, 10, so that your screen displays randint(1,50,10).  Press ENTER
      c)  Now, if you want to cause these numbers to be stored in L1, before pressing ENTER in item b,
          press STO;2nd, L₁. The  entries, randint(1,50,10)->L₁, will appear on the screen.
      d)  Press ENTER and the numbers generated will appear on the screen and will be stored in list L₁.
      e)  Alternately, you can go to the lists, hightlight the name where you want the numbers stored and then enter
           the randint(1,50,10) as described above.
 2. Calculating Factorials.
     a)  Suppose we want 5 factorial (5!).  From the home screen press 5.
     b) Press MATH and cursor over to PRB and press 4 (!)). You will have 5! pasted to the screen.
     c)  Press ENTER and you answer, 120, will be displayed.

3. Calculating Permutations.
    a)  Suppose we want the permutations (arrangements) of  8 things 3 at a time, enter 8 on the home  
         screen.
    b) Press MATH and cursor over to PRB and press 2, (nPr). You will have 8 nPr pasted to the screen.
    c)  Enter 3 and press ENTER.  You will get 336.

 4 . Calculating Combinations:.
     a)  Suppose we want the combinations (groups) of  8 things 3 at a time, enter 8 on the home screen.
     b) Press MATH and cursor over to PRB and press 3. (nCr). You will have 8 nCr pasted to the screen.
     c)  Enter 3 and press ENTER.  You will get 56
 

5. Geometric Probability Distributions:
   A. geometpdf( 
        Suppose that we want to know the probability of throwing a six on the third throw of a fair die.
        a) Press 2^nd, DISTR, ALPHA, E to display the following:

           Geometpdf
        P: 1/6
        x value: 3
        Paste
  b) Move the cursor to PASTE and press ENTER. Press ENTER again to get.1157…

B. geometcdf(:

    Suppose that we want to know the probability of throwing a six on the with no more than three throws
    of a fair die.
    a) Press 2^nd, DISTR, ALPHA, F to display the following:

       geometcdf
    P: 1/6
    x value: 3
    Paste
   b) Move the cursor to PASTE and press ENTER. Press ENTER again to get.4212…

6. Hypergeometric Probility Distriution:
    Suppose that we have 25 people from which a jury of 12 people will be chosen at random. There are 15 women
   and 10 men in the pool. Find the probability of getting 5 men and 7 women. 

NOTE: The thing that makes this a hypergeometric problem rather than a binomial problem is that the selections
  are made without replacement.

Let N= 24   (The total number of people.)
Let m = 10   (The number of men.)
Let w = 15  (The number of women.)
k = 5   (The number of men chosen.)
n= 7  (The number of women chosen.)
j = 12  (The number in the jury.)

The formula is as follows:
_{(P(k=k) =m}C_{k *w}C_n/(_NC_j)
Filling in the formula as follows:
P(k=5) = ₁₀C₅*₁₅C₇/(₂₄C₁₂)

a) Enter 10, press MATH; move the cursor to PRB and press 3 and the multiplication sign. This will give you 10 nCr 5 *
b) In the same way, enter 15 nCr 7 and the divide symbol.
c) In the same way as above, enter 24 nCr 12. Enclose this last term in parentheses. Your final set-up should be as
    follows:
₁₀C₅*₁₅C₇/(₂₄C₁₂)    (Using p(k=5)
d) Press ENTER to get .59996....

 IV. Two-variable Statistics
 1)   Scatter Plot
       First you need to get your data into lists.  
       a)  Go to the graphing screen by pressing the Y= button and deselecting any  functions so that  
             they won't be entered on your graph.  If you want to clear the lists before entering data, see the 
             note at the beginning of this document.
       b)  Press [STAT], [ENTER] to go to the list tables.
       c)  Enter the data-point numbers ( the x-values)  in L1 and the corresponding values (y-
            values)   in L2.  It is not absolutely necessary to sort your data, but the TRACE will sometimes operate
             in a confusing manner without sorting.  So, I recommend sorting.  To sort, press[STAT], select 2,   
             SortA( for ascending order.  SortA(  will be posted to the home screen.  Press 
           [2nd], L₁, 2nd, L2, [ENTER]. 
      d)  Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
      e)  Move the cursor to the scatter diagram, the first icon opposite Type,  and press [ENTER] to highlight the 
            scatter diagram icon.
      f)  Enter L₁ in the Xlist, and L₂ in the Ylist (do this by pressing 2nd and the appropriate list button);   
            then select the type marker you prefer.  (I like the + symbol. ).
      g)  Press [Zoom], 9 and the scatter plot will appear on the screen.  You can use TRACE and the arrows
            to move along and read the data pairs.

2)  Plotting  x-y line chart 
      Do that the same as the scatter plot in item 1 above except that when you select the type, choose the  
      second icon for the line symbol rather that the scatter-diagram  icon.

3)  Regression Analysis:
      Assume that you have the following information on the heights and weights on a group of young 
     women:

       First you need to get your data in lists.  You can do that from the home screen, but if you have any    
      significant amount of data, it's much easier to enter it into List tables.  See the note at the beginning of
      this document for instructions on clearing lists if you want to clear your lists before data entry.
       Here's how to enter data:
      a)  Press [STAT], [ENTER];  then enter the numbers for the independent variable, x-values,   in L1 and 
            the corresponding values in L2.
      b)  After you have finished entering data, Press[STAT].  
      c)   Cursor to CALC and press <8>, [ENTER] (Where <8> is just the number 8 from the keyboard.) 
            Note that if you want to use QuadReg or some other analysis, press the number to the left of that 
            entry. After pressing <8>, the following will appear on the home screen:
           Reg (a+bx)
           Xlist: L₁
            Ylist: L₂
            FreqList    
            Store RegEq:
            Calculate     
       d) If you want to graph the equation of the best-fit line, ship to item “e” below. If you don't want to graph, 
            continue with these instructions.   If you have your data in the L₁ and L₂ as shown in the display, move the cursor to Calculate and press [ENTER].  If you have your data in other lists, you’ll need to enter the lists by pressing 2^nd, press the appropriate list number  In either case a, b, r², and r will 
           be displayed.  Note that if r and r² are not displayed, press 2^nd , CATALOG, D, ; then scroll to DiagnosticON 
           and press ENTER. ANSWER:  If you pressed ENTER you should have these values:  a=-186.47.., b=4.705…,
           r² =.63366…, and r=.7979…
      e)  Now, you want to store the equation as a Y-variable, say, Y1, do it this way:  Press [VARS], move the cursor to 
            Y-VARS, [ENTER], [ENTER].  You should now have Y1 opposite Store RegEQ      
 f)  Move the cursor to Calculate and press[ENTER].   The values for a, b, r and r² will appear on the home screen.    
      g)  To graph the equation, you could just press GRAPH.  Depending on your data values, you may need to  
            adjust the WINDOW.  Pressing ZOOM, 0 (zero), for ZoomFit will get you a preliminary window setting.
      i)  Note that if you have already done the regression equation without storing it in a Y-variable, you  
           can do that as follows:
          1)  Press Y=; then VARS; then 5 (Statistics).
          2)  Cursor over to EQ and press 1 (or ENTER). The regression equation will be stored in the Y1=  position.  You can then graph as indicated previously.

4)  Plotting a graph with the scatter plot and the regression equation on the same axis.
     First you need to do the regression graph as described above in item 3.  Now, you want to put the  
     scatter plot on the screen with the graph. To do this:
     a) Press [2nd], [STAT PLOT] and press [ENTER], ENTER to turn Plot 1 on.
     b) Move the cursor to the scatter diagram for Type (the first icon) and press [ENTER] to highlight the scatter   
         diagram..
     c) Enter L1 in the Xlist, and L2 in the Ylist; then select the type marker you prefer. (I like a + ).
     d) Press ZOOM, 9 (for ZoomStat) and the scatter plot and best-fit graph will appear on the screen.
     e)  You can press [TRACE] to display the x-y values of the data points, or press the down arrow to  
           jump to points on the line.
Note that if your data has several decimal places and you'd rather have fewer, you can make the data
 friendlier by making the x-distance (xmax-xmin) a multiple or sub-multiple of 9.4. 

  5.  Testing the Correlation Coefficient:
        Suppose that we have the data given in the table below and we want to test the correlation coefficient at a significance level
        of 1%.  Further suppose we believe that the correlation coefficient is positive. The null hypothesis is that the correlation is 0
       

      a)  If the data are not already in the lists, press STAT, ENTER and enter the x-values in list L₁ and the 
            y-values in list L₂.
       b)  Press STAT, move the cursor to TESTS, and scroll down to LinRegTTest.  Press ENTER and the following screen will be displayed.
             LinRegTest
             Xlist: L₁
              Ylist: L₂
               Freq: 1
               β & ρ : ≠ 0   <0    >0
       c)  On the screen that appears, move the cursor down to >0 and press ENTER.  If REGEQ has an entry, delete that.
       d)  Move the cursor to Calculate and press ENTER.
       e)  Among the items displayed is P=.009218...which is less that 0.01.  So, we reject the null hypothesis
             and conclude that the correlation coefficient is positive.  

INFERENTIAL STATISTICS

V.  Normal Distribution: 
      Note:  In this section, a general method will be outlined; then a specific example will be worked.  The same
      problem will be used in several of the examples.

       General, normalcdf(:  This function returns the value of the area between two values of the random variable
          "x."  This can be interpreted as the probability that a randomly selected variable will fall within those two
          values of "x," or as a percentage of the x-values that will lie within that range.  The syntax for this function is
          normalcdf( lower bound, upper bound, μ, σ.  If the mean and standard deviation are not given, then the
          calculation defaults to the standard normal curve with a mean of 0 and a standard deviation of 1. I use the 
          values -1E9 and 1E9 for left or right tails.  The E in obtained by pressing 2nd, EE.  This can be used to solve
          such problems as the following: P(x<90), P(x>100), or P(90<x<120).  If µ and σ are omitted, the default 
             distribution allows the solution of the following: P(z<a), P(z>a), or 
          P(a<z<b).

   1.  normalcdf(: Area under a curve between two points with μ (mean) and σ (std. dev.) given.
        Suppose we want to find P(x< 20)
        a)  Press 2nd, DISTR, 2.  The following display will appear on the home screen. Note that
             the numbers have already been entered below. Enter the numbers in your calculator so that the entries are as
             indicated below:
             normalcdf
             lower: -1E9
             upper:20
             μ : 18
             σ: 5
             Paste
        b)  Scroll down to “Paste” and press ENTER, ENTER and the probability will be displayed. Notice that  you do not explicitly convert the points to z-values as in the hand method.
          Ex. 1:  Assume a normal distribution of values for which the mean is 70 and the std. dev. is 4.5. 
         Find the probability that a value is between 65 and 80, inclusive.
          a)  Complete item a) above.
          b)   Enter numbers so that your display is the following: 
                normalcdf
             lower: 65
             upper: 80
             μ : 70
             σ: 4.5
             Paste
            
          c)  Press ENTER and normalcdf(65,80,70,4.5 will be pasted to the home screen. Press ENTER again  you'll get 0.85361 which is, of course, 85.361 percent.

    2.  normalcdf(: Area under a curve to the left of a point with μ (mean) and σ (std. dev.) given.   
         Ex. 2:  In the this problem, determine the probability that the value is less than 62.
           a)  Complete item a) in the general method above.
           b)   Enter numbers so that your display is the following:
                normalcdf
             lower: -1E9
             upper: 62
             μ : 70
             σ: 4.5
             Paste
            Notice that  the "-" is a negative sign, not a minus sign.  Enter "E" by pressing 2nd, EE (The comma  key.)
           c)  Move the cursor to “Paste” and press ENTER, ENTER and you'll get 0.03772 which is, of course, 3.772 per percent.

     3.  normalcdf(: Area under a curve to the right of a point with μ (mean) and σ (std. dev.) given.
         Ex. 3:  In the above problem, determine the probability that a value is greater than or equal to 75.
           a)  Complete item a) in the general method above.
           b)   Enter numbers so that your display is the following: ( Enter "E" by pressing 2nd, EE, the comma key.)
                 normalcdf
             lower: 75
             upper: 1E9
             μ : 70
             σ: 4.5
             Paste
           c) Move the cursor to “Paste,” and press ENTER. The expression normalcdf(75, 1E9,70,4.5 will the pasted to the home screen

           c)  Press ENTER again and you'll get 0.13326 which is, of course, 13.326 per percent.

     4.  normalcdf(:  Sample taken from a normal distrubution:
          Suppose a sample of 35 is taken from the population above (μ=70 and σ=4.5).  What is the probability
          that the mean is greater than 72?
          a)  Complete item a) in the general method above.
          b)   Enter numbers so that your display is the following: (Enter "E" by pressing 2nd, EE The comma key.)
               normalcdf
             lower: 72
             upper: 1E9
             μ : 70
             σ: 4.5/√(35)  
             Paste

           c) Move the cursor to “Paste,” and press ENTER. The expression  normalcdf(72, 1E9,70,4.5/√(35) will be pasted to the home screen.
           c)  Press ENTER again and you'll get .00427... which is, of course, 0.427 per percent.

     5.   ShadeNorm(:  Displaying a graph of the area under the normal curve. 
           General:  This function draws the normal density function specified by µ and σ and shades the area
           between the upper and lower bounds.  This is essentially a graph of normalcdf(.  It will display the
          area and upper and lower bounds.  Not including µ and σ defaults to a normal curve.  The following
          instructions, "a" through "c," are general instruction to follow.
           a)  First turn off or clear any Y= functions or Plots that may be active.  Do this by moving the cursor to a 
                highlighted = sign and pressing ENTER to deselect Y= functions. For Plots, move to the
                highlighted plot and press ENTER. Clear any drawn item by pressing 2^nd, DRAW, 1, ENTER.
           b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and the following will appear on the home screen:
                
            Shadenorm
             lower: 68
             upper: 90
             μ : 70
             σ: 4.5 
             Paste
          c) Enter the correct parameters. The above entries are from example 2 above.          
          d)  Press ENTER, and the graph may be visible on the screen.  You will almost certainly need  
                to reset the Window parameters by pressing WINDOW and changing Xmin, Xmax, Ymin, and 
               Ymax settings to get a decent display. As a first approximation, set Xmin at 5 standard   
               deviations below the mean and Xmax at 5 above the mean. (See the following example.)  Start out with
               a Ymax about   0.3 and go from there.  You can set the Ymin at 0, or if you wish, set it at about
               negative one-fiftieth of Ymax.  You may need to fine tune from there. 
            Ex 1:  Draw the graph of example 2 above.
                  Be sure to clear any drawn figure by pressing 2^nd, DRAW, 1, ENTER
              a)  Press WINDOW and set Xmin=50, Xmax=90, ymin=-.01, Ymax = 0.1.  You can reset the  
                   scales as you choose to eliminate the broad baseline.  
              b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and the following will appear on the home screen:
                  Shadenorm
             lower: -1E9
             upper: 62
             μ : 70
             σ: 4.5 
             Paste
           ShadeNorm( will appear on the 
                  home screen. 
              c)  If your display does not have the entries as indicated above, enter those parameters.
              d)  Move the cursor to Draw, press ENTER and a reasonable looking graph should appear on the screen.
               Note that if you reset the window, you may need to activate the expression again.  To do that, press
               2nd, ENTER, ENTER.

       6.  invNorm(:  Inverse Probability Calculation:  
            A. Finding a z-vlaue for a particular confidence level: 
           Suppose you want the z-value for a particular α, e.g., 5%. Do this: 
           a)  Press 2nd, DISTR, 3 for invNorm(.  The following will be displayed on the home screen:
                 invNorm
                area: .05
                 µ: 0
                 σ: 1
                  Draw
           b)  Enter α for a left-tailed or 1-α for a right-tailed. In this case, Enter .05 opposite area.
           c)  Enter 0 (zero) opposite µ and 1 opposite σ if they are not already there.
           d) Move the cursor to Paste and press ENTER, ENTER. The z-value of -1.6448... will be displayed. 
           
             B. Find x that Corresponds to a Confidence Level:
                Find the number x, in a normal distribution such that a number is less than x with a given  
            probability. The syntax for this is invNorm(area, [μ, σ]).  The part in brackets indicates that there 
            is a default for those values.  The default is the standard curve with mean=0 and standard deviation is  1.
           Ex. 1:   In a distribution with a mean of 70 and standard deviation of 4.5, find the number x, such that a
                 randomly selected number will be below that number with a 90% probability.  
            a)  Press 2nd, DISTR, 3 to select invNorm(.
            b)  Enter parameters so that your display looks like this:
                invNormal
               area; .90
               μ: 70
               σ: 4.5
               Paste
               c) Move the cursor to “Paste” and press ENTER, ENTER and your answer will be 75.766.
            Ex. 2:  Given a normal distribution with a mean of 100 and standard deviation of 20.  Find a value X_o such
               that the probability is .6523 that the given x-value is below X_o.  That is P(X<X_o) = .6523.
                 a)  Press 2nd, DISTR, and fill in the display so that the entries are as follows:
               invNormal
               area; .6523
               μ: 100
               σ: 20
               Paste
              b)  Move the cursor to “Paste” and press ENTER, ENTER and your answer will be 107.83.
              Ex. 3:  What is the lowest score possible to be in the upper 10% of the class if the mean is 70 and the
                standard deviation is 12? 
                 a)  Press 2nd, DISTR, 3 and fill in the display so that the entries are as follows:
                                    invNormal
               area;1-.1,
               μ: 70
               σ: 12
               Paste
             b)  Move the cursor to “Paste” and press ENTER, ENTER and your answer will be 85.38 or 86 rounded off..

       7.   ShadeNorm(:  Window Settings for Graphing (shading) the Inverse Probability area:
            General:  If you are accustomed to graphing using the standard WINDOW settings called by 
              ZOOM, 6, then you're in for a big surprise if you use those settings for graphing the normal 
              curve.  So, before you display the ShadeNorm( function, press WINDOW and set the values 
              as follows:
               a)  Xmin = μ - 4σ. Round of to the next integer.
               b)  Add the same number to the mean that you subtracted from the Xmin to get Xmax.
               c)  Xscl= Set at the standard deviation.
               d)  Ymin=0.  Some people like to set this at a small negative number, but if you have 
                    problems with a wide range of std. devs. you'll have to keep changing it.  I set it at 0; then  
                    I'm done with it.  
               e)  Ymax= As a first approximation, set this at 0.4/σ.
                f)  Yscl= Most of the time the y-axis is not displayed, so I usually just set it at 0.01 and 
                    leave it there.

        8.  ShadeNorm(:  Graphing (shading) the Probability area:
            Ex. 1:  Obviously if you wanted to graph the example immediately above, you could use the 
            ShadeNorm( using the lower bound of -1E9 and the upper bound of 75.766.  You would do that 
            as follows:  
             a)  Press WINDOW and set Xmin=50, Xmax=90, ymin=-.005, Ymax = 0.1.  You can reset the  
                   scales as you choose to eliminate the broad baseline.  
              b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and the following display will appear:
               ShadeNorm
               lower: -1E9
               upper: 75
               μ: 70
               σ: 4.5
               Draw
            c)  Move the cursor to Draw and press ENTER and a reasonable looking graph should appear on the screen.
               Note that if you wanted to shade the region where the probability would be greater than 90%, 
               you would choose 75.766 for the lower boundary and 1E9 as the upper bound.
            
            Ex. 2:  Suppose you wanted to graph a distribution and shade the area between the points 40 and 54,
              with a mean of  46 and a std. dev. of 8.5
            a)  Press WINDOW and set Xmin=12, Xmax=80, Ymin=-.005, Ymax = 0.06.  You can reset the  
                   scales as you choose to eliminate the broad baseline.  
              b)  Press 2nd, DISTR and cursor over to DRAW.  Press 1 and the following will appear on the home screen:
                 ShadeNorm
               lower: 40
               upper: 54
               μ: 46
               σ: 8.5
               Draw
The area under the curve, 0.56562, will be displayed on the screen along with the upper and lower bounds.

        9.   normalpdf(:  Probability Distribution Function using normalpdf( :
             General:  This function can be used to find the fraction, and therefore also the percentage, of the 
                distribution that corresponds to a particular value of x.  The syntax of this function is 
                normalpdf(X, μ, σ
             A) Finding the Percentage of a Single Value:
              Ex. 1:  Suppose that the mean of a certain distribution is 60 and the standard deviation is 12.  
              What percentage of the population will have the value 50?
                 a)  Press 2nd, DISTR, 1 to display the following on the home screen. The following list
                       list of items will also appear:
                       x value:
                        μ:
                        σ:
                 b)  Enter 50 opposite "x value," 60 opposite " μ," and 12 opposite "σ." 
                 c) Move the cursor to Paste and press ENTER and the following will appear on the home screen: normalpdf(50, 60,
                    12).               
                 d)  Press ENTER and your answer should be .02317 which is about 2.3 percent.
             B)  Graphing the distribution:   
               Ex. 1:  Suppose that the mean of a certain distribution is 60 and the standard
                              deviation is 12.  
                Investigate percentages for several x-values.
                 a)  First press WINDOW and set Xmin 12 (mean minus 4 std. dev.).  Set Xmax at the same 
                      number of units above the mean, i.e., 108.
                 b)  Press Y= and select the Y1= position; then press 2nd, DISTR, 1 to paste normalpdf( to   
                      the Y1= position.
                 c)  Enter data so that the entry after Y1= looks line this:  normalpdf(X, 60,12.
                 d)  Press ZOOM, 0 to select ZoomFit and the curve should appear on the screen. 
                 e)  Press TRACE and you can move along the curve and read the values for different x-
                     values.  If you want a specific value, perhaps to get rid of the x-value decimals, just enter 
                     that number and press ENTER.

          10. ZInterval:  This function gives the range within which the population mean is expected to fall with a desired
               confidence level.  The sample size should be > 30 if the population standard devation is not 
               known.                                                                            
              Ex. 1:  Suppose we have a sample of 90 with sample mean x¯  = 15.58 and σ = 4.61.  What is the 95%
                confidence level interval?
                   a)  Press STAT, cursor to TESTS, and press 7:ZInterval. The following will be displayed.
                         ZInterval
                       Input: Data  Stats
                       σ: 4.61,
                       x¯ :15.58,
                       n:90
                       C-Level: .95
                   b)  Move the cursor to "Stats" on the ZInterval screen and press ENTER.
                   c)  If the data is not as appears above, enter the correct data opposite positions as indicated in
                         the display above.
                  d)  Finally, move the cursor down to Calculate, press ENTER, and the interval (14.628, 16.532) will appear along with
                        the values for "n" and the mean.
              Ex. 2:  Suppose that you have a set of 35 temperature measurements with population σ = .5 and you want 
                         to know with a 95% confidence level within what limits the population mean of temperature measurement 
                         will fall.
                   a)  First you need to enter the data in a list, say L_1, by pressing STAT, ENTER, and entering your data
                        in the list that appears.  Just enter a data point and press either ENTER or the down arrow.
                   b)  Press STAT, move the cursor to TEST and press 7 to get the ZInterval screen. The following will be displayed:
                       ZInterval
                    Input: Data Stats
                    σ: 0.5
                    List: L₁
                    Freq: 1
                    C-Level: .95
                     Calculate
                   c) First move the cursor to "Data" and press ENTER.
                   d) Now enter the value as indicated above if they are not already entered. Enter .5 opposite σ:.
                   e)  Next you need to have the sample mean. That is obtained by entering the list name where the data
                        is stored opposite List. Press 2nd, L₁ , or whatever list you have your data in.
                    f)  Enter additional information as follows:  Freq: 1, C-Level: .95.
                   g)  Move the cursor to Calculate and press ENTER.  The same type data will be displayed as in Ex. 1 above.

VI. Other Distributions and Calculations:

         1. TInterval:  If the sample size is <30, then the sample mean cannot be used for the population mean, and ZInterval
              cannot be used.  However, if the distribution is essentially normal, i.e., known to be normal form other sources or
              has only one mode and is essentially symmetrical, then the Student t Distribution can be used.
              Ex. 1:  Suppose you take ten temperature measurements with sample mean x¯  = 98.44 and s = .3.
                What is the 95% confidence level interval?
                   a)  Press STAT; move the cursor to TESTS, and press 8:TInterval.
                   b)  On the screen that appears, cursor to "Stats" and press ENTER.
                   c)  Enter data opposite positions as follows:  x¯ :98.44, S _x : .3   n:10, and C-Level: .95.
                   d)  Cursor down to "Calculate", press ENTER, and, after a few seconds, the interval (98.225, 98.655)
                        will appear along with the values for  "n" and the mean.
              Ex. 2:  Suppose that you have a set of 10 temperature measurements and you want to know with a 95% 
                         confidence level what limits the population mean of temperature measurement will fall within.
                   a)  First you need to enter the data in a list, say L_1, by pressing STAT, ENTER, and entering your data
                        in the list that appears.  Just enter a data point and press either ENTER or the down arrow.
                   b)  Press STAT, cursor to "TEST" and press 8 to get the TInterval screen.
                   c)  Cursor to "Data" on the TInterval screen and press ENTER.
                   d)  Enter information as follows:  List: Press 2nd, L₁, Freq: 1, C-Level: .95.
                   e)  Cursor to "Calculate" and press ENTER.  After a few seconds, the interval (xx.xxx, xx.xx)
                        will appear along with the values for  "n," the mean, and sample standard deviation.

            2.  Student's t Distribution:  The Student's t Distribution is applied similar to the normal probability function, but it
                 can be applied when there are less than 30 data points, for example: P(t> 1.4|df = 19).  The last part means
                 that the number of degrees of freedom (one less that the number of data points) is 19.
               Ex. 1:  Find the probability that t> 1.4 given that you have 20 data points.  
                 a)  Press 2nd, DISTR, and press 6 for tcdf.
                 b) Enter data so that your display appears as follows: (Except for the note.)
                      lower: 1.4
                      upper: 1E9     (The “E” is obtained by pressing 2^nd, and the comma key.)
                      df: 19
                      Paste
                 c)  Move the cursor to Paste and press ENTER, ENTER and your answer should be .0888...

           3.  invT: Finding a t-value Given α and df:
                a)  Press 2nd, DISTR, 4, and the following will be pasted to the home screen:
                       invT(
                     area:
                      df:
                      Paste
                 b) Opposite "area," E=enter α or 1-α, depending on whether you have a left or right tail; then enter the
                     degrees of freedom opposite df.
                 c)  Move the cursor to Paste and press ENTER and the value for "t" will be displayed.  Note that you may
                      need to divide α by 2 if you have not already made that adjustment.  
            4.  The Chi-squared Distribution:  The χ² Distribution is implemented similar to the Student's t 
                 Distribution.  
                Ex. 1:  Assume that you want to find P(χ² > 24|df=20) the same as in the above Student's t Distribution. 
                 a)  Press 2nd, DISTR, 8 to paste χ²cdf( to the home screen.
                 b)  Enter data so that your display is as follows:
                    χ²cdf
                   lower: 24
                   upper: 1E9
                    df:19
                    Paste
                 c) Move the cursor to Paste and press ENTER, ENTER and your answer should be .1961...

          5.  Binomial Distribution, binonpdf(:
             Suppose that you know that 5% of the bolts coming out of a factory are defective.  You take a sample of 12.  
             Determine the probability that 4 of them are defective.
                a)  Press 2ND, DISTR, move the cursor down to A:binompdf( and press ENTER.
                b)  Enter numbers so that your entry is like this:
                     binompdf
                   trials: 12
                   p: .05
                   x value: 4.
                   Paste
                c) Move the cursor to Pate and press ENTER, ENTER and 0.00205… will be displayed.

               6.  Binomial Distribution, binoncdf(:
               Suppose that you know that 5% of the bolts coming out of a factory are defective.  You take a sample of 12.  
               Determine the probability that 4 or more of them are defective.
               First I'll show a very easy way that gives only the answer; then I'll show a method that takes more time, but 
               provides much more intermediate results.
                Short Way:
                a)  Press 1, and then - , the subtraction sign.
                b)  Press 2ND, DISTR, move the cursor down to B:binomcdf( ( or alternately press ALPHA, B) and press ENTER.
                c)  Enter numbers so that the display looks like this: 
                     binomcdf
                    trials: 12
                    p: .05
                     x value:  3
                     Paste
                d)  Press ENTER. ENTER and the answer, .0022364 will be displayed.
                 Longer Way:
                  This method will give you the individual values for 0 through 3 as well as the sum
                  of all of those values.
                  First press STAT, ENTER and clear list L₁.
                 a) Press 2ND, DISTR; then move the cursor to A:binompdf( (or press ALPHA, A) and press ENTER.
                 b)  Enter information so that your display looks like this: 
                       binompdf
                      trials: 12
                       p:.05
                       x value:  {4, 5, 6, 7,8,9,10,11,12}.    (Be sure to use braces rather than parentheses.)
                       Paste
                   c) Press ENTER and the  expression binompdf(12, .05, {4, 5, 6, 7,8,9,10,11,12}) will appear on the home screen.
                  d)  Press STO, 2ND, L₁ to tell the calculator which list to store the individual values in.  
                   e)  Press ALPHA, : (the decimal point key); then 2ND, LIST, move the cursor to MATH, and press 5. : sum( should now  be displayed on the home screen. 
                   e)  Press 2ND, L_1,.  You should now have this expression:  binompdf(12, .05, {4, 5, 6, 7,8,9,10,11,12}) : sum( L1). 
                   f)  Press ENTER,  and the answer, .0022364, will be displayed.  The individual numbers,  are in list L₁.  Just press STAT, ENTER to see them.
                    Ex 2:  Suppose in the above example you want to know the probability of 3 and fewer.
                      a)  Press 2ND, DISTR, move the cursor down to B:binomcdf( and press ENTER.
                      b)  Enter numbers so that the display looks like this: 
                          binomcdf
                      trials: 12
                       p:.05
                       x value: 3
                       Paste
                     c)  Move the cursor to Paste, press ENTER, ENTER and the answer, .997763... will be displayed.
                   Ex 3:  Suppose that, on average, one out of ten apples in a fruit stand is unacceptable.  What is the probability that
                          8, 9, or 10 of a set of 11 such apples are acceptable?
                      a)  Press 2ND, LIST; move the cursor to MATH and press 5 to paste sum( to the home screen.
                      b)  Press 2ND, DISTR, ALPHA, A. 
                      c)  Enter data so that you have sum(
                            binomialpdf
                            trials: 11
                            p .9
                            x value:  {8,9,10}  ( Be sure to use braces rather than parentheses.)
                            paste
                      d)  Press ENTER and the following expression should appear on the home screen: sum(binomialpdf(11,.9,{8,9,10})
                            should appear on the home screen.
                       e) Press Enter and .667...will be displayed.

VII.  Hypothesis Testing:
      1.  Testing for Mean of z Distribution with Statistics:  
           a)  Press STAT and move the cursor over to TESTS.
           b)  Press 1 or ENTER for Z-Test.
           c)  Move the cursor to Stats and press ENTER.
           d)  Opposite µ_o, enter the mean for the null hypothesis.
           e)  Enter the given values for σ, x-bar, and n.
           f)  Select the proper condition for the alternative hypothesis. 
           g)  Move the cursor to Calculate and press ENTER.  The z-value, p-value and some other statistics will 
                be displayed.     

      2.  Testing Mean  of z Distribution with Data:
           a)  Enter the data into L₁ or whatever list you choose.
           b)  Press STAT and move the cursor over to TESTS.
           c)  Press 1 or ENTER for Z-Test.
           d)  Move the cursor to Data and press ENTER.
           e)  Opposite µ_o, enter the mean for the null hypothesis.
            f)  Opposite σ, if you are using the sample standard deviation and it is not given, do the following: Press 2nd,
                LIST, move the cursor to MATH and press 7. The term stdDev(, will now be displayed opposite σ.  Now, enter your
                list number where the data is stored by pressing 2nd, and the list number, for example L_1 .  Note that you do not
                divide stdDev by √n.
           g)  Enter L₁ opposite List and 1 opposite Freq.
           h)  Select the proper condition for the alternative hypothesis. 
            i)  Move the cursor to Calculate and press ENTER.
            j)  If you want to use the calculator to find the z-value or critical value, see those procedures below.

      3)  Finding a z-vlaue for a particular confidence level: 
           Suppose you want the z-value for a particular α, e.g., 5%. Do this: 
           a)  Press 2nd, DISTR, 3 to display the invNorm screem as follows:
                  invNorm
                area: .05
                µ: 0
             α: 1  
           b)  Enter α for a left-tailed or 1-α for a right-tailed. In this case, Enter .05 opposite area.
           c)  Enter 0 (zero) opposite µ and 1 opposite σ if they are not already there.
           d) Move the cursor to Paste and press ENTER, ENTER. The z-value of -1.6448... will be displayed. 

       4)  Finding critical values of x.  
           Suppose you have a mean of 5.25, standard deviation of .6 and you want the critical number for an α
               of 5%.  
             a)  Press 2nd, DISTR, 3, and invNorm screen will appear. ( will be pasted to the home screen.
             b)  Enter numbers so that your entry looks like this: 
                  invNorm
                area: .05
                µ: 5.25
                σ: .6
                Paste
               The entry .05 is the value for α for a left tail. For a right tail enter 1-α which would be .95 in this   
                    problem.
             c) Move the cursor to Paste and press ENTER, ENTER and the value 4.26 will be displayed.  

       5.  Testing for Mean  and t Distribution with Data:
           a)  Enter the data into L₁ or whatever list you choose.
           b)  Press STAT and move the cursor over to TESTS.
           c)  Press 2 for T-Test. The following screen will be displayed:
                   T-Test
               Input:Data   Stats
               µ_o:
               List:
               Freq:
               µ: ≠µ_o < µ_o   >µ_o
               Calculate    Draw
           d)  Move the cursor to Data and press ENTER.
           e)  Opposite µ_o, enter the mean for the null hypothesis.
           f)  Enter L₁ opposite List and 1 opposite Freq.
           g)  Select the proper condition for the alternative hypothesis. 
            h)  Move the cursor to Calculate and press ENTER.
            i)  If you are working a problem using the p-value test, read the p-value and compare it with α or α-1 as appropriate.
            j) If you are working a problem using the t-value test, you will need to know the critical values for the level of 
               significance, α, that you have chosen.  There are different options depending on your needs. See "invT: Finding a
               t-value Given α and df:" in section VII of this document for the details of these options.
                

       6.  Testing for Mean and T Distribution with Statistics:  
           a)  Press STAT and move the cursor over to TESTS.
           b)  Press 2 or ENTER for T-Test. The following screen will be displayed:
                   T-Test
               Input: Data   Stats
               µ_o:
               List:
               Freq:
               µ: ≠µ_o < µ_o   >µ_o
           c)  Move the cursor to Stat and press ENTER.
           d)  Opposite µ_o, enter the mean for the null hypothesis.  
           e)  Enter the given values for σ, x-bar, and n. If you don't know x-bar you can enter it by placing the cursor opposite
                the symbol for mean; then press 2nd, LIST, cursor to MATH, and press 3; then press ENTER. Enter L₁ and
                press ENTER.
           h)  Select the proper condition for the alternative hypothesis. 
            i)  Move the cursor to Calculate and press ENTER.
            j)  If you are working a problem using the p-value test,  read the p-value and compare it with α or α-1 as appropriate.
            k) If you are working a problem using the t-value test, you will need to know the critical values for the level of 
               significance, α, that you have chosen.  There are different options depending on your needs and whether 
               you're using a TI-83 Plus or a TI-84 Silver Edition. See "invT: Finding a t-value Given α and df:" in section VII of
               this document for the details of these options.
          7. InverseT:
             a) Press 2nd, DISTR, 4 and the following screen will be displayed:
                   invT
                   area: .25
                   df: 12
                   Paste
              b) Press ENTER. If the numbers above are entered the following will be displayed: inv(.25, 12)
              c) Press ENTER again and -.695... will be displayed.      

VIII.  Statistics of two Populations:   
      1.  Confidence Interval for Two Dependent Populations: 
  Enter the data from population 1 into L₁ and the data from population 2 into L₂.  Do this as follows:
     a)  Press STAT, ENTER, and enter the data in the displayed lists.
     b) After entering the data, highlight the list name, L₃ example, where you want to store the data.
   Now, store the paired differences in list L3 as follows:
     c)  Press 2nd, L1, minus sign, 2nd, L2.
    d)  You should now have L1 - L2 at the bottom  on the lists screen.  Press ENTER and the differences
           will be stored in list L₃.
  Now, find the confidence level as follows:
     e)  Press STAT, move the cursor to TESTS, and press 8 for TInterval. The following screen will appear:
             TInterval
           Input: Data   Stats
           List: L₁
           Freq: 1
           C- Level: .95
           Calculate
     f)   On the screen that appears, move the cursor to "Data" and press ENTER; then enter the list name opposite
           List and 1 opposite Freq 
    g)   Enter the confidence level you want opposite C-Level, for example .95.
    h)  Move the cursor down to “Calculate” and press ENTER.  The confidence interval and other statistics will be 
             displayed.   

2.  Confidence Interval for Two Dependent Populations (Stats):
      If you do not have data, but have the mean, standard deviation, and n, use this procedure.
     a)  Press STAT, move the cursor to TESTS, and press 8 for TInterval.
     b)   On the screen that appears, move the cursor to "Stats" and press ENTER. The following screen will
            appear:
             TInterval
           Input: Data   Stats
           x:                                      (Should be x with a bar over it.)
           S_x:
           n:
           C-Level:
           Calculate
     c)  Enter the sample mean, standard deviation, and then the number of data points opposite "n." 
     d)   Enter the confidence level you want opposite C-Level, for example .95.
      f)  Move the cursor down to “Calculate” and press ENTER.  The confidence interval and other statistics will be 
             displayed.   TO HERE 9/30 $$

 3.  Confidence Interval for Two Independent Populations (Stats): 
       a)  Press STAT, move the cursor to TESTS, and press 0 (zero).
       b)  On the screen that appears, move the cursor to Stats and press ENTER.
       c)  Enter the sample means, standard deviations, and number of data points, n, for each sample.
       d)  Set the confidence level you choose opposite "C-Level."
       e) Highlight "No" opposite "Pooled" if there are no assumptions about the variations.
       f)  Move the cursor to "Calculate" and press ENTER.  The confidence interval along with other statistics will be  
           displayed.

4.  Confidence Interval for Two Independent Populations (Data): 
        Enter the data from population 1 into L₁ and the data from population 2 into L₂.  Do this as follows:
     a)  Press STAT, ENTER, and enter the data in the displayed lists.
           To go to the confidence interval screen do this:
      b)  Press STAT, move the cursor to TESTS, and press 0 (zero).
      c)  On the screen that appears, move the cursor to Data and press ENTER.
       d)  Opposite "List 1," press 2nd, L₁ and opposite "List2," press 2nd, L₂.
       e)  Set the confidence level you choose opposite "C-Level."
       f)   Highlight "No" opposite "Pooled" if  there are no assumptions about the variations.
        g)   Move the cursor to "Calculate" and press ENTER.  The confidence interval along with other statistics will be  
              displayed.

5. Standard Error of Estimate, S_e:
     Suppose that we have two variables x, and y withe the relationships listed in the following table:    

Let's calculate the standard error of estimate, "s" on the TI-84.
        a) Press STAT, ENTER and enter the x-values in list L1 and the y-values in list L2.
        b) Press STAT and move the cursor over to TESTS.
        c) Press ALPHA, F; then move the cursor to Calculate and press ENTER.
        d) You should get a value of 3.91... opposite "s" in the display.
        NOTE: If you want the statistic for the slope, beta, that is also available as t = 3.805....

IX.  Other Tests and Inferences:

       1.  One-way ANOVA:
            Suppose that you are trying to determine a better way to motivate learning and come up with the following scores in response
      to different types of motivations:
     

    Determine if  one of the methods is better that the others..  
         The null hypothesis,  H_o, will be that all of the means are equal.  Suppose that we want to determine if they are by 
         99 % confidence level.  
   a)  Press STAT, ENTER, and enter the each group of data in lists L₁ through L₄. To clear old data from a list, place the cursor on 
         the list title, press CLEAR; then ENTER.  DO NOT press DEL to clear a list. 
   b)  The syntax for ANOVA is ANOVA(List 1, List 2, List 3, List 4, ...List n).  So,  press STAT, move the cursor to TESTS, and move
         the cursor down that list to ANOVA(.  That's item H: on the TI-84.  Press ENTER and ANOVA( will be displayed on the home 
         screen. 
     c)  Inter list numbers so that you have the following display: ANOVA(L₁, L₂, L₃, L₄.  You can either close the parentheses or not.
    d)  Press ENTER and various results will be displayed.  One of these is P=.004764....  So, H_o is rejected. 

2. Chi-Square Test for Independence:
     Suppose that we have the observed (those indicated by O= ) values in the following table and we want to know if independence
    is indicated at the α=0.01 level.
   

     First we need to calculate the expected values, i.e., those already entered as E= in the table.  The expected value is this:
   P(cell) * Sample Size. 
   The probability, P, of a cell is calculated using the rows and column totals.  For example the probability cell 5 is as follows:
   P(5) = P(brand 3) and P(Group 1)
           =40/200 * 80/200
   Now, we multiply that by the sample size of 200:
   E=40/200 * 80/200*200
      =16
  This can be simplified to the following:
  E=[(Row Total)(Column Total)]/(Sample Size)
  Now we want to test for independence, and we will first enter the observed values in Matrix [A] and the expected values in Matrix [B].
  a)  Press 2ND, MATRIX, move the cursor to EDIT and press ENTER to edit matrix [A].
  b)  Enter 4 x 2 for the matrix configuration and then enter the observed values.  Press 2ND, QUIT to leave this matrix.
  c)  Press 2ND, MATRIX and move the cursor to EDIT.  Then press 2 to edit matrix [B].
  d)  Enter 4 x 2 for the matrix configuration and then enter the  expected values in the matrix.  Press 2ND, QUIT to end the 
        matrix editing.
  e)  Press STAT,  move the cursor to TEST, and select Χ² -Test from that list and press ENTER.
  f)  The Calculate screen will be displayed with the matrices indicated for storing observed and expected.
  g)  Move the cursor to Calculate  and press ENTER.  Χ² =24.64 and p=1.79467 will be displayed.  
  h)  Since P is smaller than the α-value of 0.01, we reject the null hypothesis of independence and conclude that the preferences are dependent.
  i)  Alternatively, we could have compared the statistic of Χ=24.64 with the critical value for α=0.01.  Using a df = (4-1)(2-1) and consulting a 
        table, we would have found Χ² =11.34.  Since the test statistic is larger, we would reach the same conclusion as above.

3. Х² Goodness of Fit:
     The TI-83 Plus does not have a GOF function, so I will first give the procedure for the  TI-84; then I will give a procedure for the 
     TI-83 Plus.    
     Suppose that a cell phone vendor wants to test the colors of the cases of cell phones to see if customers have a color preference.  
    A sample is taken and the data in the following table is collected.  The vendor wants a confidence level of 95%.  That is,  α =.05.

  The expected values are calculated by adding all of the observed values and dividing by 5, the number of categories.
  H_O: Customers show no color preference.
  H_1:  Customers show a color preference.
   a) Press STAT and enter the observed values in L₁ and the expected values in L₂. 
   b) Press STAT, move the cursor to TESTS, and cursor down to D: Х² GOF – TEST and press ENTER.     
       Alternately, you can press ALPHA, D to activate that procedure.
  c)  On the screen that appears, make sure that L₁ is opposite Observed and L₂ is opposite Expected. 
 d)  Enter 4 opposite df.  The value of df is one less than the number of categories.
  e)  Move the cursor to Calculate and press ENTER.  
   f)  The value P= .01882... will be displayed.  Since this value is less than α=.05, we reject the null   
        hypothesis.

Part III: DOING PROCEDURES BY HAND WITH THE AID OF A CALCULATOR  

X.  Aids in doing statistics by hand.
       General:  Often in book problems in school you'll need to do a lot of calculations by hand.  These   
         techniques will save you a lot of arithmetic. But first a word about lists. Lists are a powerful tool for doing statistics. In most computation
formulas, lists can be substituted for the variables in most applications.   When doing list arithmetic such as 
multiplication, addition, subtraction, and raising to a power and storing the result in another list, the operation 
can be done from either the list screen or the home screen.  For example L₁*L₂ with the L₃ title highlighted will 
do the same thing at the list screen as L₁*L₂→L₃ at the home screen.  (The arrow is a result of pressing STO.)  
There may be occasions when a function such as sum( must be done from the home screen, but it usually more 
convenient to do arithmetic operations from the lists screen. But when using a function such as sum( , the operation 
must be done from the home screen. 

 
 1.  Arranging Data In Order.  (This is the same as item 2 in section I above, which I will repeat here.)
     a)  Enter the data in one of the lists as indicated in Section I.     
     b)  Press STAT, 2 (SortA).  This will paste SortA( to the home screen.  
     c)  Press 2nd, L₁ (or whatever list you want to sort); then press ENTER.  "Done" will be displayed
          on the home screen, indicating your data has been sorted. Note that you can also sort data in  
          descending order with SortD.  If you want to sort data in an independent and dependent list, L₁ and L2_, 
             for example, use SortA(L₁,L₂).

2.  Finding Mean (x-bar), ∑x, or ∑x² , σ, Median, Q₁, Q₃ for Grouped or Ungrouped Data.
    For Ungrouped Data:
    a)  After entering your data in the list as described in item 1 of Section I, above, press STAT, and 
         cursor over to CALC, and press ENTER. "1-Var Stats" will be pasted to the home screen. 
    b)  Move the cursor opposite List and enter the list name you want to operate on by pressing 2nd; then the list number, for example L_1.
     c) Move the cursor to Calculate and press ENTER.
    d)  A number of results will be displayed on the home screen. 
     NOTE:  You can also find these values for discrete random variable statistics by entering the values
                 of the variable in L₁, for example, and the corresponding data values in L₂. 
   For Grouped data:
    a) Find the midpoints of each group and enter those values in L₁. You can have the calculator
do the arithmetic by entering the two values divided by 2. Finally, enter the corresponding frequencies in
        L₂.  Entering data in a list is described in item 1 of Section I, above.
         
    b)  Press STAT, cursor over to CALC, and press ENTER. The "1-Var Stats" screen will be pasted to the home screen. 
     c)  Move the cursor opposite List and press 2nd, L₁; then move the cursor opposite FreqList and press 2nd, L₂. Finally move the cursor to calculate and press ENTER.
    d)  Various statistics will be displayed on the home screen.  Note that for grouped data, ∑xf is listed on the
          calculator as ∑x and ∑x² f is listed as ∑x² .

3.  Finding results such as xy or (x-y):
     a) Assume that your x-data is in L₁ and your y-data is in L₂.  Then obtain the product by pressing 
        2nd, L₁; x (multiply symbol), 2nd, L₂, ENTER.
     b)  If you want the data stored in a list, L₃ for example, first press STATS, ENTER and highlight the list name L₃.
          Now, press 2nd,  L₁, x (Multiply symbol),  2nd, L_2.  Then press ENTER.
     c)  Obviously, x-y can be obtained by merely substituting the subtraction symbol for the 
          multiplication symbol in the steps above.

4.  Squaring operations such as elements of lists.
     a)  To square the elements of a data set, first enter the data in a list, for example L₁.
     b)  Press 2nd, L₁; then the x² symbol, ENTER.  The squared elements will be displayed. 
     c)  If you want to store the squared data in a list, for example L₃, first press STATS, ENTER and highlight the list name L₃.
          Now, press 2nd,  L_1, x² (the square symbol; then press ENTER.
     d)  If you want to multiply corresponding elements of two lists and square each result; then your 
         expression should be like this:  (L₁ * L₂)² .

5.  Find x-x¯ (Sorry, I have no symbol for the mean, so I displaced the bar.) from the data in 
     list  L₁ and store the results in L_2.
     a)  Enter 2nd, L₁, -.  Note that" -" is a minus sign not a negative sign.
     b)  Cursor to MATH and press 3.  You should now have "L₁-mean(" pasted to the home screen.
     c)  Press 2nd, L₁, ENTER.  The result will be displayed on the home screen.  
     d)  If you want to store the results in a list, for example L₃, highlight the list name where you
          want the data stored; then enter the operation as described above.  Finally, press ENTER.
        

 6.  Finding (x-x¯ )²  
      a) Move the cursor to the title for L₂ and  press (, 2nd, L₁, -. Note that the “-“ is a minus sign.
      b)  Press 2^nd, STAT, move the cursor to MATH and press 3.  You should now have "(L₁-mean(" pasted to the home screen.
        c)  Press 2nd, L₁,),),x² .  The expression (L₁-mean(L₁))² should now be displayed on the screen.
          Press ENTER and the results will be displayed in L₂.
      Note that this can also be done from the home screen if desired. 
 

7.  Finding (Σx)² and Σx²
     Some computation formulas for the standard deviation require (Σx)² .  To find that, do the following:
      a)  Enter your data in a list as described at the beginning of this document.  Press 2nd, QUIT to get 
          out of the list. Press ( to enter a parenthesis on the home screen.
      b)  Press 2nd, LIST, and cursor over to MATH.
      c)  Press 5.  "(sum(" should be entered on the home screen.
      d)  Press 2nd, L₁ or whatever list your data is stored in.
      e)  Press ), ), x² .  You now should have (sum(L₁))² on your home screen.
       f)  Press ENTER and the results will be displayed on the screen.
       g) Σx² can be found by using the "1-Var Stats" function under STATS, CALC, but you can also 
          find it by entering "sum L₁² "

8.  Notice that you may also do several other operations by pressing 2nd, STAT; then moving the cursor to 
    MATH and entering the list name that you wish to operate on.

XI.  Plotting a by Hand:
      1. Histogram by Hand:
             First make a suitable table for Class Limits, Class Boundaries, and Frequency to record the numbers
               that will be found in the procedure that follows.   
            Determine the class limits and the number of data points in each class as follows:
                             a) Go to the lists by pressing [STAT], [ENTER] and entering your data in L₁ or whatever
                     list you choose as indicated in Entering Data at the beginning of the procedure.

                  
               b)  You will need to enter the class width. It is best to determine that at this point. Do that as follows:
                    First you will need to sort the data so that you can determine the smallest and largest number.
                    (1) Press STAT, 2 (for SortA) to paste SortA( to the home screen. Press 2^nd, L_1. You should now have SortA(L₁
                          on the home screen.
                    (2) Press STAT, ENTER and record the smallest number (the first one) and the largest number (the last one).
                    (3) Now calculate the number of data points using the following method:
                         ( Largest number – smallest number)/ (number of desired classes). Round to the
                         next higher integer. NOTE:  It may also be that you are given the number of classes by a textbook problem. 
                         In that case simply enter that number is place of "C" in the above formula.  
               c)  Press [2nd], [STAT PLOT] and press [ENTER], [ENTER] to turn Plot 1 on.
                 d)  Move the cursor to the icons opposite Type, select the third icon, histogram, and press [ENTER] to  
                       highlight the histogram icon.
                    f)  Enter L₁ (or whatever list your data is in) opposite Xlist, by pressing 2nd, L_1.  Make sure there's a
                      1 opposite Freq if you have ungrouped data. You may need to press ALPHA to toggle the mode from
                       alphabetical to numerical to enter a number.
                    g)  Press [ZOOM]; then 9 (ZoomStat) and the histogram will appear on the screen.
                  h)  To get the number of classes you want, press WINDOW and change Xscl to the class width you calculated
                        above
               i)  Press GRAPH and the new graph with the correct number of classes will be displayed. 
                 j) If you want the class boundaries rather than the class limits, Press WINDOW, enter    -
                    -0.5 after the value for X-min and press ENTER. A new value for X-min will be 
                    displayed. Press Graph to plot the histogram with the boundaries.
               k)  To get the numbers for plotting, press TRACE and move the cursor across the tops bars of the graph using
                    the cursor controls. The class limits will be 
                 displayed. Write these down in your table.  If you have a display such as min=1, max<9, the class limits of that class would 
                 be 1 and 8. Don't forget that the lower class limit is counted as part of the class width.  The number of data points in the 
                 class being viewed will be indicated by n= some number. 
              b)  Determine the class boundaries as follows:
                   First, you must set the lowest class boundary. If, for example, the lower class limit is 1, press WINDOW and change x-min 
                   from 1 to 0.5. Press GRAPH to redraw the histogram. Now press TRACE and move the cursor across the top of the histogram 
                   bars.  The numbers displayed are the class boundaries.  Consider the < symbol as an equal sign for the upper boundary.  
                   Record these number in the table that you made.
              c)  You now have all of the data you will need to draw the histogram by hand. 

  2) Box and Whisker Plot by Hand
           You can save yourself considerable calculation if you use the calculator to find Q₁, Median, and Q₃ 
           when doing a box-and-whisker plot by hand.  To find those values do the following:
           a) First enter your data; then press STAT, cursor to CALC  and press ENTER.  The "1-Var Stats" screen will be
               displayed on the home screen.
          b)  If your data is in list L₁ move the cursor opposite List and press 2^nd, L₁; then move the
cursor to Calculate and press ENTER. 
          c)  The values for Xmin, Xmax, Q₁, Q₃, and Med will be displayed. That’s all you need to
                for the box and whisker plot. 

   XII. Statistics Using Calculation Intensive Techniques:
     1)  Doing a Discrete Probability Distribution by Hand
          Many teachers still see value in cranking out the numbers for these statistics, so here are methods
              to take some of the drudgery out of doing the arithmetic. 
            The mean can be obtained by the following formula: mean = Σxp(x).
            To obtain the individual values and store them in list L₃, do the following:  (The x-values should
             should be stored in L₁  and the p(x) values in L₂.)
             a)  From the list screen, highlight the title of L₃ and press 2ND, L₁, x, 2ND, L₂..  You will now have L₁*L₂
                   at the bottom left of the list screen. 
             b)  Press ENTER and you will have the individual values stored in list L_3.
             c)  To get the sum of these values,  do this.
                    CAUTION:  DO NOT store sums in the lists if the particular list is going to be used in a succeeding arithmetic 
                    operation.  Instead, do these calculations from the home screen. 
                      (1)  Move the cursor down to the first blank space in L₃.   Press 2nd, LIST; cursor to MATH, and press 5.  
                             The expression sum( will be displayed at the bottom of  the list screen.  
                      (2)  Press 2ND, L₃ , ).You will have sum(L₃) at the bottom of the list screen. 
                      (3)  Press ENTER and the sum of those values will be displayed as the last item of L₃.  
             You can obtain the variance and standard deviation by first solving for the variance using the formula:  Σx² P(x) - µ² 
              where µ is the mean obtained as above.  To obtain the individual values of the first term,  x² P(x), and store them in list L₄, 
              do the following:
             a)  From the list screen, place the cursor on the title for list L₄ , press 2ND, L₁, x², ,x, 2ND, L₂.  You will 
                   have L₁²*L₂ at the bottom left of the lists.
             b)  Press ENTER and the individual values will be entered in list L_4. 
             c)  To get the sum of these values do the following:
                      (1)  Caution:  Do not store sums in lists if you plan to use this list in another arithmetic operation.  Place the cursor 
                             in the first blank space in L₄, then press 2nd, LIST; cursor to MATH, and press 5.  The expression 
                             sum( will be displayed at the bottom left of the LIST screen. 
                      (2)  Press 2ND, L_4, )..  You will have sum(L₄) at the bottom of the list screen.  
                      (3)  Press ENTER and the sum of those values in L₄ will be displayed as the last entry in L_4..  
              d)  Now we want to subtract the value for µ² from the last value obtained and that will be the variance.  You can always do that
                     by hand but if you want to be a little more creative, do it this way.  First press 2ND, QUIT to go to the home screen. 
                     Suppose that your sum for L_3, µ, and L₄, Σx² P(x), are in rows 6.   Press 2ND, L_4, (, 6, ), -, 2ND, L_3, (, 6, ).  You should now have this:
                      L_4,(6 ) -L_3,( 6).  Press ENTER and the variance will be displayed.
               e)  To calculate the standard deviation from the variance in the list assuming that the variance is in L₃(7),  move the cursor down one
                     space to L₃₍8)and press 2ND, √, 2nd, L₃,(,7,) and press ENTER.  The standard deviation will be displayed in L₃(8).
               f)  Of course if you calculated the standard deviation from the home screen, if you have just calculated the variance, press 2ND, √, 
                    2nd, ANS, ENTER.  
            NOTE:   Obviously, if you only want to obtain the values for the  these three parameters,  you can
            use this method, but it is much easier to let the calculator do it all.   Just as information, the total
           expression for the

Standard Deviation

  B.  Calculating Numbers to Plug into a Computation Formula::
     The standard deviation can be found easily by using 1-Var Stats as described above, but  
      many teachers require that students do the calculations themselves to learn the details of the  
      process.  The following gives a method for using the TI-84 for doing much  
     of the arithmetic required and obtaining numbers to plug into the formulas.
     Suppose that students did sit-ups according the table shown below.    

The variance computation formula is as follows:   s² = [(Σx² -(Σx)²)/n)]/(n-1), where s² is the variance . 
So,  we will need x² ,  ∑x² and ∑x to plug into the formula.
   a)  Enter the data in the table as indicated previously in this document. 
   b) Press STAT, move the cursor to CALC, and press ENTER. The “1-Var Stats” screen should be
       pasted to the home screen. Enter information so that the appearance is as follows:
       1-VarStats
      List: L₁
      FreqList:
     Calculate
   Move the cursor to Calculate and press ENTER. The numbers listed below will be displayed.
   c)  Copy n=7, ∑x = 216, and ∑x² =7492 and Sx = 11.73923.
  NOTE:  You now have enough data to plug into the formula and solve for the variance and standard deviation. 
 If you are not required to show the detailed calculations, skip to filling in the formula in step “f.”  Otherwise, continue
 with the next step.
   d)  Now we’ll need an x² column.   Place the cursor on the title L₂, press 2^nd, L₁, x², ENTER.  The squares of the
          numbers in L₁ will be displayed in L₂.  You can enter into your table  the numbers that you found for n, ∑x, and 
         ∑x² from the 1-Var Stats function.
     e)  Now, we want to use the numbers that we previously recorded to plug into the variance 
        formula.   So, from the home screen enter (7492-216²/7)/(6).  You can either merely enter these numbers or your
          worksheet or test sheet and square the standard deviation you found above and enter for the answer, or you can 
           do more time and work to enter the numbers in your calculator and find the variance.
     f)  If you entered the numbers in the calculator, press ENTER and you should have 137.8…, which is the variance.
     g)  To find  the standard deviation, press 2ND, √ , 2ND, Ans, ENTER,  or you can just record the standard deviation that
           and you recorded above.  In either case, you will have 11.73...

 B.  Calculating Standard Deviation from Grouped Data to Plug into a Computation Formula:
     The standard deviation and variance for grouped data are similar to ungrouped data except that the
     x-values are replaced by the midpoints of the classes.  Let's assume some sort of grouped  
     data as indicated by the first and third columns below. 
    

The formula for the grouped data variance is this:
           s² =( Σx²  -(Σxf)² /Σf)/(Σf-a)  
a) You can either do the midpoints by hand  and store them in L₁ or calculate and store them in list L₁ as follows:
      (1) Store the lower boundaries in list L₁ and the upper boundaries in L₂.  
      (2) Place the cursor on the title of L₁; then press (, 2ND, L₁, + 2ND, L₂,), ÷, 
                  2 .  You should have (L₁ + L₂)/2 at the bottom left of the tables.  Press     
                  ENTER and the midpoints will be stored in L₁.
 Now let’s calculate the required numbers.
b) Press STAT, move the cursor to CALC, and press ENTER. The expression “1-Var Stats” 
    should be pasted to the home screen along with the following list:
      List:
      FreqList:
     Calculate
   You must enter the list where the data is stored opposite List. Press 2^nd and the key number for 
   the list where the data is stored.
c) Move the cursor opposite FreqList and enter the frequencies opposite the appropriate numbers  
    in an adjacent list. For example, if the data is in L₁; then enter the frequencies in L₂.
 d) Move the cursor to highlight "Calculate" and press ENTER.  The standard
     deviation and several other statistics will be displayed.
e) Record the standard deviation, Sx =14.868 for a reference.  Also record ∑x=∑xf=3780, 
     ∑x²=∑x²f=296600, and n=50.  You’ll need these values later.  
      Notice that the value for ∑f is listed as n in the calculator and ∑xf is listed as ∑x and ∑x²f is   
      listed as ∑x².  
NOTE:   You now have enough numbers to plug into the formula and solve for the variance.   
If you are not required to do the detailed calculations to fill in the table, skip to item “j” below.  
Otherwise continue with the next step from the Lists screen. .    
d)  Calculate xf and store it in L₃ by placing the cursor over the title for L₃ pressing  2ND, L₁, *, 2ND, L₂._. 
       You should have L_1*L₃ at the bottom left of the tables.  Press ENTER and the products will be stored 
       in list L_3.
e)  Calculate x²f by placing the cursor on the title for L₄  and  pressing 2ND, L_1, x² , * , 2ND,  L₂.  
     You should now have L₁² *L₂  at the bottom left of the tables.  
f)  Press ENTER and the results will be stored in list L_4. 
 g)  You don’t need to calculate Σf.  That is the value for “n” that you previously recorded.   
 h)   You don’t need to calculate Σxf.  That is the value for ∑ x that you previously recorded.
 i)  At this point you can either just record the formula with the numbers plugged in on your work sheet
     or test sheet or you can do the extra work to do the calculation with your calculator.  To get the
     answer without putting the numbers in your calculator, merely square the standard deviation, which
     you previously recorded. If you're going to do it with the calculator, do the next steps.
     Now, you want to plug the appropriate numbers into the formula for the variance. From the  
     home screen enter (296600-3780²/50)/(49)
 j)  Press ENTER and you should have 221.06, which is the variance.
 k)  If you want the standard deviation, you can just use the one you previously recorded or you can calculate
       calculate by pressing 2ND, √ , 2ND, Ans, ENTER, and you will have 14.868...
      Note that if you calculated the standard deviation first, just square that value to get the variance.

2)  Finding the Values a and b for the Best-Fit Equation Using a Computation Formula:
      We will use calculator functions to reduce the arithmetic necessary for these formulas.  First we will
      use "2-Var Stats" to obtain the values for such expressions as ΣxΣy and Σx² which we can enter in the
      formula on our worksheet or test sheet.  Then, we will use the calculator function LinReg(a+bx to find
      the values for "a" and "b" without doing the arithmetic on our calculators.  Finally, for those who are 
      allowed to use a simper method than the arithmetic intensive formula, I will suggest the use of matrices 
     and the calculator function rref( for finding the final answers. intensive substitution method used in many textbooks.

Assume that you have the following information on the heights and weights on a group of young women:   

     The formula for “b” is this:  (nΣxy –ΣxΣy)/(nΣx²- (Σx)²).  So, you will need to record the values 
      for x-bar, y-bar, Σx, Σy, ΣxΣy, Σx², Σy², and n.. You can get all of these by  using the 2-Var Stats function.   
     Use that as  follows:
      a)  With the data in lists L₁ and L₂ press STAT, move the cursor to CALC, and press 2.  The 
           expression 2-Var Stats, should be displayed on the screen.  
      b)  Press ENTER and the necessary values will be displayed.  Notice that you will need to 
           scroll down to get some of the values on the screen.  Record the values for the following 
          parameters:  x¯=65.125, Σx=521,  Σx²=33979, 
          n=8, Σy=960, y-bar=120, Σy²=116900, Σxy=62750
       At this point you will save yourself a lot of time if you calculate "a" and" b" with the calculator.  To do that, 
       press STAT, move the cursor to CALC and press 8 for LinReg(a+bx. The following screen will appear.
             LinReg(a +bx)
        XList:L_{1
          YList:}L_{2
        FreqList:
          StoreRgEQ:
          Calculate}
         Move the cursor to Calculate and press ENTER and values are displayed for several summations that are
        needed for the formulas.
       If you are required to show your work, you will need
      to write the numbers and on your paper.  
      c)  Plug these numbers into the formula and then enter the expression in your calculator.  
           Just a few notes on entering the data in the calculator:  All denominators and numerators 
          with more than one term must be enclosed in parentheses.  On theTI-84, be sure to press the right arrow
          when finished with your entries. That will end the entries under the square root sign.
          Method I: Enter the numbers into the formula by hand.
    d)  Enter the following  formula on your worksheet.
         b=(nΣxy –ΣxΣy)/(nΣx²- (Σx)²). 
          Now record these numbers on your worksheet or in the calculator  if you're going to calculate the values.: 
           =(8*62750-521*960)/(8*33979-521²)
           =4.7058…..
     e)  Now, record this formula on your worksheet:  
          a= y-bar –b(x-bar)
          Now just enter these numbers on your worksheet or enter them in your calculator if you're going to calculate the value.
            =120-4.7058 *65.12
            =-186.465… 
          Method II: Transfer the numbers from VARS to a Matrix:
          After you use LinReg(a+bx), the numbers that you need are stored in VARS. So, we will transfer the numbers to
          matrix [A] and solve the matrix for "a" and "b."
a. After you have set up the matrix as described in Method I, go to the home screen by pressing 2^nd, QUIT.
b. To enter the value 6, for “n,” in the first element of the matrix press 6, STO, 2^nd, MATRIX, ENTER, (1, 1). You
should have 6à[A](1,1) on the screen. Press ENTER to transfer the value to the matrix.
c. Now, we want to do the summations. First press 2^nd, ENTER to display the expression above once more. Place the
cursor on the value 6 and press VARS, 5. Move the cursor to ∑ and press ENTER.. Move the cursor to the element designation and change it to 1,2. You should now have ∑x à[A](1,2). Press ENTER to display the value and transfer it to the matrix.
d. Press 2^nd, ENTER to display the expression again and change the matrix element to 2,1. You should now have ∑x à[A](2,1). Press ENTER to display the value and transfer it to the matrix.
e. Now, we want to transfer ∑y in matrix element 1,3. First press 2^nd, ENTER to display the expression above once more.
Edit the expression using VARS, 5; move the cursor to ∑and press 3. Change the matrix element so that you have ∑yà[A](1,3). Press ENTER to transfer the number to the
matrix.
f. Now, we want to transfer ∑x² into matrix element 2,2. Display the expression once again and edit it using the listings under VARS, Statistics to get ∑x²à[A](2,2). Press ENTER to transfer the information.  Using the same technique, First press 2^nd, ENTER to display the expression above once more.
g. In a similar manner, transfer the ∑xy to element 2,3.  You should have ∑xy à[A](2,3).
h. Now we want to solve the matrix. Press 2^nd, MATRIX, move the cursor, to MATH and scroll down to rref(.
That’s usually item “B.” Press ENTER to display rref( on the home screen.
i. Now you want to tell the calculator which matrix you want to solve. To do that, press 2^nd, MATRIX, ENTER if
you have the data in [A].
j. Finally, press ENTER to display the answer matrix. It should be the following

┌ 1 0 .4ךּ
└ 0 1 1 ﻠ
Sorry, my effort at making matrix symbols leaves a lot to be desired.

3)  Finding the Correlation Values r and r² Using a Computation Formula:   
      We will use calculator functions to reduce the arithmetic necessary for these formulas.  First we will
      use "2-Var Stats" to obtain the values for such expressions as ΣxΣy and Σx² to enter on our worksheet
      or test sheet.   Then we will use LinReg(a+bx to find the values for "r" and "r²."  
      This last procedure will eliminate the necessity for entering the numbers for the formulas into our calculators 
      to get the final answers.
     
Assume that you have the following information on the heights and weights on a group of young women:

       First you need to get your data in lists.  You can do that from the home screen, but if you have any    
      significant amount of data, it's much easier to enter it into List tables.  See the note at the beginning of
      this document for instructions on clearing lists if you want to clear your lists before data entry.
       Here's how to enter data:
      a)  Press [STAT], [ENTER];  then enter the numbers for the independent variable, x-values,   in L1 and 
            the corresponding y-values in L2.
            NOTE:  The formula for “r” is this:  (nΣxy –ΣxΣy)/[(√nΣx²- (Σx)²)(√nΣy²- (Σy)²)].  So, you will  
            need  Σx, Σy, ΣxΣy, Σx², Σy^2,, and n.  You can get all of these by using the 2-Var Stats 
            function.  Use that as follows:
      b)  With the data in lists L₁ and L₂ press STAT, move the cursor to CALC, and press 2.  The 
            2-Var Stats screen wills be displayed on the screen as follows:
          2-VarStats
          Xlist: L₁
          Ylist: L2
           FreqList:
           Calculate
           If the correct lists are not present, enter them by pressing 2nd and the correct list button.       
      c) Move the cursor to Calculate and press ENTER and the necessary values will be displayed.  If the  
          Notice that you will need to scroll down to get some of the values on the screen.  Record the 
          values for these parameters:  Σx=521,  Σx²=33979, n=8, Σy=960, Σy²=116900, Σxy=62750.
      NOTE:  Just a few words on entering the data in the calculator:  All denominators and numerators
      with  more than one term must be enclosed in parentheses. To end a square root  expression on the
     TI-84,  press the right arrow.
      Now let’s plug the numbers into the equation for r: 
       At this point you will save yourself a lot of time if you let the  calculator calculate r and r².  To do that, 
       press STAT, move the cursor to CALC and press 8 for LinReg(a+bx.  Enter information so that the display
       is as follows:
         LinReg(a +bx)
      Xlist: L₁
      Ylist: L2
      FreqList:
       Calculate
       Move the cursorto Calculate and press ENTER. The values for r and r² will be displayed along with other
       statistics.  If you are required to show your work, you will need to write the numbers and on your paper.  
       Record the following:
        d)  r= (nΣxy –ΣxΣy)/[(√(nΣx²- (Σx)²)(√(nΣy²- (Σy)²)]
                = (8*62750-521*960)/(√(8*33979-521²)(√(8*116900-960²))
               =.7979…..
           If you chose to put the numbers in the calculator, you might want to read the following:
       e)  Some students seem to have difficulty accurately entering a long expression such as in item "d."  
            Those students can do the calculation without loss of accuracy by using the following method.
          1)  Enter the numerator in the calculator and store it in variable N.  In this manner:  
               8*62750-521*960, STO, ALPHA, N.  
          2)  Calculate the denominator and store it in two separate variables M and D. In this manner
               √(8*33979-521² )  , STO, ALPHA, M; then √(8*116900-960²), STO, ALPHA, D         . 
          3)  N÷(M*D), ENTER.  You'll get the same answer as above. 
        g)  Of course, r² is just the square of Ans, or you can just copy if from the stats calculation.  

XII:  Appendix:
 Program for sorting data into classes:
NOSCAL
:FKIZER090210
:SortA(L₁)
:min(L₁)-->S
:dim(L₁)-->Q
:max(L₁)-->M
:int(M/W)+1-->dim(L₂)
:Input "CLS WDTH  ",W
:0-->T:1-->X:W-->F:0-->C
:ClrHome
:Lbl 1
:While L₁(X)≥S and L₁≤F
:T+1-->T
:X+1-->X
:If X>Q
:Then
:T-->L₂(C+1)
:Goto 2
:End
:End
:C+1-->C
:T-->L₂(C)
:0-->T
:S+W-->S
:F+W-->F
:Goto 1
:Lbl 2
:L₂
After you’ve entered the program, use it in this manner.
a)  First enter the data in list L_1. The data need not be in any order.   
b)  To execute the program, highlight the program name and press ENTER.         
c) The program will ask for the class width, CLS WDTH.  Enter the class width and press ENTER.
d)  The numbers for the classes will be stored in list L₂ and that list will be displayed after execution.  Note that you can
       move the numbers after the ellipses (the three dots) with the cursor arrows.  When finished press CLEAR to stop the

 

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