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Brief User Guide for TI-89 Titanium Statistics

This document covers single- and two-variable statistics, scatter plot, regression analysis, normalpdf, Student's Distribution, hypothesis
testing, statistics of two populations, and much more.

INDEX:

To facilitate lookup, the instructions are divided into the following categories:

         I.   Data Manipulation - Entering data, sorting data, using the APPS screen, friendly values from graphs,
       
II.  Single-Variable Statistics - Histogram by hand, simple histogram with the calculator,  frequency polygon
             from ungrouped data, cumulative frequency (Ogive) graph, Relative frequency polygon and Cumulative relative
             frequency Graphs, histogram using grouped data, frequency polygon using grouped data, cumulative frequency
            from grouped data, relative frequency polygon and cumulative relative frequency graph from grouped data,    
            
percentile graph, box and whisker, box and whisker with outliers, box and whisker by hand,  discrete probability 
             distribution, discrete probability distribution by hand.
       III.  Two Variable Statistics scatter plot, regression analysis, scatter plot and regression analysis on same
             axis,
       IV.  Aids in doing statistics by hand - Putting data in order, finding mean,
Σx, Σx2, σ, median, Q1, Q3, finding
             products such as xy and finding x-y, squaring elements of lists, finding x-xbar, finding (x-xbar),
 Σ(x-xbar),
           
 finding  Σ(x)2 or  Σx2  .
        V.  Permutations, combinations, factorials, random numbers, generating a random data set, selecting
             numbers randomly from a normal data set,
       VI.  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(,
ZInterval,
       VII. Other Distributions -  TInterval, invT Finding a t-value given
α and df, Chi-squared Distribution
      VIII.  Hypothesis testing - mean and z-test (data), mean and z-test (statistics), mean and t-test (data),
              mean and t-test (statistics).
        IX. 
Statistics of two Populations - confidence interval for two dependent population, confidence interval for two 
               independent populations (Data and Stats),

      

IMPORTANT NOTICE:  This is a "beta" version, which means that I have only typed it out without actually
      checking out the procedures on the calculator.  It should be reasonably accurate, but if you find errors, please
      send me an e-mail.  To do that, just click on  "E-Mail Webmaster" in the navigation bar and an e-mail form
      will appear.  I will do some more editing soon after the fall semester starts.

RELEASE DATE:  Not Released        DATE LAST REVISED:  4/2/13
A beta version of the  in printer friendly format appears here.

NOTE:  See copy restrictions and printing hints at the end of this document.  

Preliminaries:  We will be using lists a lot in these procedures.  You can assign your own variables to a list, but in
  these procedures, I will be using list1 through list6.  For simplicity, I will call this the list tables. Many, perhaps most,
  of these procedures will be done from the list screen, but some operations, especially arithmetic operations, will be
  done from the Home screen.  There are several ways to paste list variables either to the Home screen or to a dialog box. 
  Here are a couple:
      a)  To enter list1 press 2ND, ALPHA, l, i, s, t, ALPHA, 1.  That takes me about 12 seconds.
      b)  To enter list1, press 2ND, VAR-Link, l (that's L not 1), ENTER.   That takes me about 8 seconds. So guess what,
            I usually use the last one.  There are others which you can look into at your leisure. 

I.  Data Manipulation
     (NOTE:  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, list1 -list6,  is to place the cursor on the name above the list and press CLEAR; then ENTER. 
      1)  Entering Data:
          a)  To call up the list tables press APPS, select the Stats/List icon and press ENTER. Either the tables will
               appear or you'll get a dialog box.  If you get the dialog box just press ENTER to go to the tables.
          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.  To delete an entry
              just place the cursor over the data and press the backspace key, ← .     
      2)  Putting Data in Order:
           
Go to the lists following the instructions in Entering Data procedure above.
           a)  Place the cursor in the list you want to sort and press F3, 2, 1.  On the dialog box that appears, the name of the
                 list you selected should be entered.  If it is not, enter it now by pressing 2ND, VAR-LINK, l, (L, not 1); then cursor
                down to the list name you want and press ENTER
           b)  Move the cursor down opposite Sort Order and select either "Ascending" or "Descending" as you prefer by
                pressing the right cursor key if you need to change the entry.
           c)  Press ENTER and the list will be sorted.  You may need to press ENTER again.

     3)  Using the APPS screen:
          a)  Press APPS to go to the screen that has icons for all of the applications. 
          b)  If the application that you want  is not on the first screen,  press the first letter of the title of the icon you want to  
             select.  For example, if you want to select Matrices press D, the first letter of the title
             Data/Matrices.  For Lists you would press  S, for "Stats/Lists." 
          c)  Press 2ND, APPS to toggle back and forth between the present screen and the previous APPS screen.  Note
              that non-APPS screens may not be not recalled by this method. 
          d)  Another way to organize the APPS functions is to place them in folders.  As for myself, I have placed the
                six APPS that I use most often in the MATH folder.  That way when I press APPS, I can then press
                F1,2, 4 and my most-often-used APPS will immediately be displayed on the screen.  To organize the APPS, do
                this:
                1)  Press APPS, F1, right cursor arrow, 3.  A dialog box with a list of APPS will appear on the screen.  Scroll
                     up or down the list with the cursor arrows and press the cursor right arrow to select one of the APPS. 
                     Frankly, for most people, I think this is more trouble than it is worth.

     4)  Friendly Values on Graphs Using TRACE:
          Many times when you use the TRACE function with a graph, you may get an x-value such as 2.784532.  If you
          change the x-min and x-max in the WINDOW function to be  7.9 or multiples of that number and the y-min and y-max to multiples
          of 3.8 or multiples, the displayed values will be "friendlier."  That is, they will be integers or numbers with one or two decimal
          places.  You can always set the values by hand,  but the easiest method is to use the ZoomDec function of ZOOM. 
          Just press F2; then 4, for ZoomDec.  If you want the more of the graph on the screen, press Zoom, set the cursor to x=0 and y=0 and press
          3 for ZoomOut.  If you want to reduce the covered area, press ZoomIn in a similar manner..
          If youre trying to find the value at a specific point, say x=0,  and the cursor does not fall on the x-axis,  you could use the zero function.
         To do that, press F5, 2.  That will set you up for   finding a zero. Select the lower bound (left); then the upper
         bound and press ENTER. Remember that any time you want to get back to the decimal window just  press F2, 4.

     II.  Single-Variable Statistics  
    
1)  Doing a Frequency Distribution Histogram by Hand:
           a)  Use items 1, 2, and 3 in Section I above to enter and sort your data.
           b)  Find the class width as follows:
                  (1)  Let S represent the smallest data number (The first number in your sorted list.), L be 
                        the largest number (The largest number in the sorted list.), and C be the number of
                        classes you've chosen. Find the class width, W, with the formula W = (L-S)/C.

          
c)  Determine the limits of the classes by adding the class width to each successive class. 
                Don't forget that the lower class limit is counted as part of the class width.
           d)  Determine the number of data points in each class as follows:
                (1)  If your data is in list1, go to that list. Make sure your data is sorted in ascending order;  
                      then scroll down to the last number that falls within the upper limit of the first class.  At
                      the bottom of the list (if you are using list1) your will see list1[#], where # is the number of data items
                      in your first class. 
                (2)  Scroll down to the last item of the second class and subtract the number of items in the
                      first class from the number that appears in list1 (#). Continue this until you come to the
                      end of the list.  Note that if you also want cumulative frequency, just write down the
                      numbers as you progress. 
            e)  Subtract 0.5 from each lower class limit of the first class to get the lower boundary of the
                 first class.  Add the class width to get successive boundaries.
             f)  Alternatively, you could do the histogram described below and use the data classes and
                values from that histogram.

      2)  Doing a Histogram with the TI-89 Titanium:
          
This procedure describes how to do  a simple histogram for which the user  selects the class (bucket)
           width and, therefore, the number of classes.  
First you need to get your data into lists. 
           a)  First go to the graphing screen by pressing ♦, F1 and deselect or clear  any  functions or plots so that they won't be
                displayed with your graph.
                Now, go to the lists and enter data as follows:
           b)  Press [APPS], select the "Stats/List" icon and press [ENTER].  If the Stats/List icon is not on the screen, you
                 can either search for it by using the cursor arrows or press S to display the icons whose titles start
                 with an "S."  Select the "Stats/Lists" icon and then press ENTER
           c)  The list tables may be displayed, if not and your list tables are in the main folder, press ENTER.  If the lists
                are not in the main folder, select the correct folder and press ENTER. 
           d)  Enter the data points in list1;  then press F2, 1 to select "Plots Setup."
           e)  Press F1 and on the dialog box that appears, select Histogram if it is not already displayed.  Do that by pressing
                the right cursor arrow and then pressing 4. 
           f)  Move the cursor down to the box opposite "x" and enter the name of the list that your data for the histogram is in.
                 Do that with this sequence of keystrokes:  2ND, VAR-LINK  (the minus key), l (that's the letter L, not the number 1)
                 and press ENTER if your data is in list1.  If the data is in  another list, highlight that list and  press ENTER.
          g)  Move to the box opposite "Hist. Bucket Width" and enter the number for your class width.
          h)   Make sure that "Use Freq. and Categories" is set to NO and press ENTER.
           i)   Press F5 to display the histogram.  You may want to change the window settings.  Press
♦, F2 to change the
             WINDOW settings. 
At a minimum, I usually  change y-min to 0, but that is a matter of personal choice. You
                 may want to use F3  (Trace) to find the maximum y-value for your graph.  You will also notice that the calculator
                 has set the class boundaries.  I usually set my own boundaries by setting x-min at 0.5 less than the smallest number, 
                 but that again is a matter of the practices of your text or teacher.  To get back to the histogram after setting the
                WINDOW, press 2ND, APPS.

          j)  To display  the numbers for the boundaries of the classes and the number of items in the class, press
                F3 (Trace); then use  the cursor to move across the tops of the bars in the histogram and read the  
                various values for max, min, and n.

     3)  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 to get the midpoints and record those values.   Store these
                values in a list,  for example list2  if you have your histogram data in list1.  Store the corresponding values of "n"
                in list3.
           c)  Press F2, ENTER, F1, press the right cursor arrow; then 2 to select xyline.
          d)  Enter list2 opposite x and list3 opposite y.  You can do this using either of the methods I have previously described.
           e)  Press ENTER and maybe ENTER again if the plot list screen does not appear.
           f)  Press F5 at the Plot Setup screen, 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 list2.  To do that place the cursor at the first item in list2, press 2ND, INS and replace
               the zero that appears with your the first midpoint you calculated. Go to the bottom of the list2 list and enter the
               second value you calculated.
         B.  Now you want to enter zero in list3 opposite each of these new midpoints.  Place the cursor at the top of list3 and press
              2ND, 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 list2.
         C. Press ♦, GRAPH and the graph will be displayed. 

     4.  Constructing a Cumulative Frequency Chart (Ogive) Graph:
           
Suppose that you have the frequencies  in list3.
          a)  Enter the x-max values that you recorded above in a list.  For example, list4  if you still have data in the  other  lists.
               If you want a return to zero graph you will need to calculate new maximum value for the beginning and end of
               end of this list.
          b)  Now, store the cumulative frequency data in list5  as follows:  Press Home, 2ND, MATH, 3, 7  to paste cumSum(
               to the Home  screen.  Note that if you do not want a return-to-zero graph, you should delete the zeros at the top and
               bottom of this new list
          c)  Press 2ND, VAR-LINK,  l (L, not 1).  Scroll to list3 or whatever list you have the frequencies store in.  Press ENTER
               to paste that list to the Home  screen.   
          d)  Close the parentheses and press STO; then  2ND, VAR-LINK,  l (L, not 1).
          e)  Scroll to list5,  or whatever list you want the cumulative frequencies stored in, and press ENTER.  The cumulative 
               frequencies will now be stored in list5.
          f)  Now you want to graph list4 as" x"  and list5 as "y."
          g)  Go to the list screen by pressing 2ND, APPS,  or by pressing APPS, selecting the Stats/Lists icon, and pressing ENTER.
          h)  From the list screen,  F2, ENTER, F1,  press the right arrow;  then 2 to select xyline.
          i)  Enter list4 opposite "x" and list5 opposite "y."  You can either type the lists in the dialog box or use VAR-LINK as
               I have described previously. 
           j)  Press ENTER and maybe ENTER again if the plot list screen does not appear.
           k)  When the plot screen appears, deselect all plots except the one of interest by pressing F4. 
           l)  Press F5 and the graph will appear on the screen.
          NOTE:  Some teachers or texts prefer return-to-zero graphs.  If your course requires that, and you have not already
          taken care of that in the steps above, do the following after step b)   above:
          A.  Calculate an x-max for a new class preceding the first class.  This value will be entered into list4.  To do that place
                the cursor at the first item in list4, press INS and replace the zero that appears with the  x-max you calculated.
         B.  Now you want to enter zero in list5 opposite this new x-max value.  Place the cursor at the top of list5 and press
              INS.  A zero will be added. 
         C. Press ♦, GRAPH and the graph will be displayed. 
       m)  If the graph does not appear on the screen, press ZOOM, 9 and the graph will appear on the screen.

     5)   Relative Frequency polygon and Cumulative Relative Frequency (Ogive) Graphs: 
             Do these exactly  as in the frequency polygon and cumulative frequency graph above except that after storing
             the data (step a) for the frequency polygon, do these steps: 
             a)  Press Home, 2ND, VAR-LINK,  move to list3, or  wherever you have the frequency data stored, and press
                  ENTER.  The term list3 should now be displayed on the home screen.
             b)  Press , N,  STO, 2ND, VAR-LINK, select list3 and press ENTER.  This will replace  the data in list3  to with
                  relative frequency.   Note that "N" is a number equal to the total number of data points that are in your sample.
             c)  Press ENTER and the procedure will be executed.  Now finish the procedure as in the frequency polygon.
                  Notice that you will need to set y-max to a number smaller than 1. 

     6)  Histogram Using Grouped Data:
  
        a)  First go to the graphing screen by pressing ♦, F1 and deselect or clear any  functions or plots so  that they will
                not  be displayed with your graph.
                Now, go to the list and enter data as follows:
           b)  Press [APPS], select the "Stats/List" icon and press[ENTER].  If the Stats/List icon is not on the screen, you
                 can either search for it by using the cursor arrows or press  S to display the icons whose titles start with an
                "S."  Then press ENTER
           c)  The list tables may be displayed, if not and your list tables are in the main folder, press ENTER.  If the lists
                are not in the main folder, select the correct folder and press ENTER.
          d) 
Enter the midpoints of the classes into list1  and the corresponding frequencies into list2, or whatever lists
                 you choose
.  Press F2, 1 to select "Plots Setup."
           e)  On the dialog box that appears, select Histogram if it is not already displayed.  Do that by pressing the right
                cursor arrow and pressing 4. 
           f)  Move the cursor down to the box opposite "x" and enter the name of the list that your data for the histogram is in.
               Do that with this sequence of keystrokes:  2ND, Var-Link (the minus key), l (that's the letter L, not the number 1)
              and press ENTER if your data is in list1.  If the data is in another list, highlight that list and  press ENTER.
          g)  Move to the box opposite "Hist. Bucket Width" and enter the number for your class width.
          h)   Make sure that "Use Freq. and Categories" is set to YES, and enter the list where the frequencies are stored
                 opposite Freq. Press ENTER, and maybe ENTER again if the Plot Screen does not appear.
           i)   Press F5 to display the histogram.  You may want to change the window settings.  At a minimum, I usually
                 change y-min to 0, but that is a matter of personal choice. You may want to use F3  (Trace) to find the maximum
                 y-value for your graph.  You will also notice that the calculator has set the class boundaries.  I usually set my
                 own boundaries by setting x-min at 0.5 less than the smallest number, but that again is a matter of the practices
                 of your text or teacher. 

          j)  To display  the numbers for the boundaries of the classes and the number of items in the class, press
                F3 (Trace);  then use  the cursor to move across the tops of the bars in the histogram and read the  
                various numbers.

   7)  Frequency Polygon Using Grouped Data:
          
Do this exactly like the histogram, except select the 2 (xyline) rather than Histogram.   If you've already done the   
           histogram,  just change "Plot Type"  and press ♦, F3. 

     8)  Cumulative Frequency (Ogive) Graph from Grouped Date:
           a)  Enter the upper class limits in a list, for example, list3  if you have data in the first two lists.
           b)  Enter the frequencies in list2  if it is not already there.  Now,  do the following: 
                A)   From the list  screen highlight the title of list4, press 2ND, MATH, 3, 7.  The expression  cumSum(  will be posted to 
                       the area at the bottom of the list screen. 
                B)  With the cursor after the parenthesis, press 2ND,  VAR-LINK, L, cursor to  list 2 and press ENTER.  Close the
                       parentheses and press,STO, 2ND, VAR-LINK, L, cursor to list4 and press ENTER .  You will now have
                      cumSum(list2)  pasted at  the bottom of the  screen.  Press ENTER.  The cumulative frequencies will now be
                      in list4.  Now, you want to plot the upper class limits as "x" and the cumulative frequencies as "y."
          c)  Press F2, 1 to select "Plots Setup."  Select the plot number you want with the cursor arrows.
          d)  From the list screen, press F1 (Define) and change the "Plot Type" to xyline
on  the dialog box that appears.
               Do that by pressing the right cursor arrow and pressing 2.
           f)  Move the cursor down to the box opposite "x" and enter the name of the list that your data for the midpoints is in.
               Do that with this sequence of keystrokes:  2ND, VAR-LINK  (the minus key), l (that's the letter L, not the number 1)
              and press ENTER if your data is in list1.  If the data is in another list, highlight that list and  press ENTER.
          g)  Move to the box opposite "Hist. Bucket Width" and enter the number for your class width.
          h)   Make sure that "Use Freq. and Categories" is set to YES, and enter the list where the frequencies are stored
                 opposite y. Press ENTER, and maybe ENTER again if the Plot Screen does not appear.
           i)   Press F5 to display the frequency polygon.  You may want to change the window settings.  At a minimum,
                 I usually change y-min to 0, but that is a matter of personal choice.

     9)  Relative Frequency and Cumulative Relative Frequency Graphs for Grouped Data:
           
Do these exactly  as in the frequency polygon and cumulative frequency graph above except that  after storing
            the data for the frequency polygon do this step: 
            a)  Press 2ND, VAR-LINK,  move the cursor to list4 and press  ENTER.
            b)  Close the parentheses and press 2ND, VAR-LINK, move the cursor to list4 and press ENTER. 
            c)  Press , N, STO, 2ND, VAR-LINK,  move the cursor to list4. Note that "N" is the number  for the quantity of data,
                 points not the letter.)
            d)  Press ENTER and you should have list4/N→list4 pasted to the home screen.
            e)  Press ENTER and the data in list4  will be converted to relative frequency. This assumes that the frequency
                 data is stored in list4 .
           f)  Continue with the plotting as in the cumulative frequency graph above. 

     10)  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 list1 and the corresponding frequencies in list2.  If you want the graph to start
                  at zero, enter the first lower boundary with zero for the frequency.
           b)  From the Home screen, press 2ND, MATH, 3, 7.  The term cumSum( will be pasted to the Home screen.
           c)  Press 2ND, VAR-LINK, l (L not 1), cursor to list2 and press ENTER.  Press ), .  You now should have
                 cumSum(list2)/ on the home screen.
           d)  Press 2ND, MATH, 3, 6.  You should now have cumSum(list2)/Sum( on the Home screen.  
           e)  Press 2ND, VAR-LINK, L, cursor to  list2, and press ENTER.  Close the parentheses.  You now should have 
                cumSum(list2)/Sum(list2) on the home screen.
           f)  Press x (the multiply symbol), 100, STO, 2ND, VAR-LINK, l, cursor to list3, and press ENTER.  You now should
                have  cumSum(list2)/Sum(list2) *100→list3 pasted to the home screen. 
           g)  Press ENTER and the data will be stored in list3 .
          Plotting the Data:
            a)  Now you want to graph list1 as x and list2  as y, so press F2, ENTER, F1, 2 to select xyline.
           b)  Enter list1 opposite x and list2 opposite y.  You can do this using either of the methods I have previously
               described.
           c)  Press ENTER and maybe ENTER again if the plot list screen does not appear.
           d)  When the plot screen appears, deselect all plots except the one of interest by pressing F4. 
           e)  Press F5 and the graph will appear on the screen.
            f)  You can find the exact percentiles of the boundaries by using Trace (F3),  and approximate percentiles of
                other x-values by using the cursor.
        
    11) Box and Whisker Plot
         a)  First go to the graphing screen by pressing the ♦, F1. .  Deselect or clear any  Y= functions or plots 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 APPS, S, highlight the Stats/Data icon and press ENTER  to go to the list tables.
         c)  Enter your  numbers in list1.  (Or whatever list you choose.)
         d)  Press F2, 1, F1, press the right cursor arrow, press 3 for Box Plot. 
         e)  Opposite "x" enter list1 or whatever list your data is in.  To do that press 2ND, VAR-LINK, l (that's L, not 1).
               If your data is in list1 press ENTER, otherwise scroll to the list your data is in and press ENTER.
         g)  Make sure "Use Freq. and Categories" is NO; press ENTER and the Plot Screen should appear.  If not, press
               ENTER again.
         h)  Press F5 and the box-and-whisker plot will appear on the screen.

         i)  To find the numbers for the limits of the quartiles, press F3 (Trace); then use  the cursor to move
               across the diagram and obtain the values for quartiles or the beginning and ending values.
             

12) Box and Whisker Plot with Outliers:
      
 You may  have one or two outliers (numbers much larger than the rest) and you may not want to include those
         data plots in your box and whisker graph because they will distort the graph.  You can do a modified box plot
         as follows:  
             a)  Do the plot exactly as in the Box and Whisker above except in step d), select 5 (Mod Box Plot) instead of
                   Box Plot.
             b)  This will not include the outliers in the last whisker, but will plot them as separate points
                    after the end of the last whisker.
 
 13) 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)  Press APPS, S and select the Stats/List icon; then press ENTER.  This will take you to the tabulated list
             screen.  Enter your data in a list, for example, list1.
       b)  Press F4, 1.  This will take you to the 1-Var Stats dialog box. 
       c)  Opposite "List" enter the list number where your data is stored, for example list1.  You can either type in
            the name or press 2ND, VAR-LINK, l (L, not 1), and press enter when the correct list number is highlighted. 
       d)  Type 1 in the box opposite Freq. and press ENTER.  If the statistic do not appear after a few seconds, press ENTER again.
       e)  Cursor down and you will find Q 1 , Q3 , and Med, Max and other statistics.
       f)  Draw the box plot using your usual method and using these numbers. 
 
14)  Discrete Probability Distribution
       
Let's take a simple example to demonstrate this:  Suppose a word is flashed on a screen several
       times while people are  trying to recognize the word.  The list below indicates what percentage of the
       group required a given number of flashes to recognize the word.  
              No. of Flashes     1        2        3     4       5
              Percent                27      31     18     9      15
              P(x)                     .27     .31     .18    .09   .15
         In summary, the method is to enter the number of flashes into list1  and the corresponding P(x)  
         values into list2  as the frequency.  The details are as follows:
         a)   Enter the number of flashes in list1 and the corresponding P(x) values in list2 opposite the
                number of flashes.  (How to enter data in a list is covered at the beginning of this document.)
        b)  From the tabulated list screen, press F4, 1.  This will take you to the 1-Var Stats dialog box. 
        c)  Opposite "List" enter the list number where your data is stored, for example list1.  You can either type in
            the name or press 2ND, VAR-LINK, l (L, not 1), and press enter when the correct list number is highlighted. 
        d)  Enter list2 in the box opposite Freq. and press ENTER.  If the statistic do not appear, press ENTER again.
         e)  If you need the variance, merely re-enter the value for the  standard deviation, σx  , and square it 

  15)  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 x-values  should be stored in list1  and
            the p(x) values in list2.)
         The mean can be obtained by the following formula: mean = Σxp(x).
            To obtain the individual values and store them in list3, do the following after storing the data :
             a)  From the Home screen press 2ND, VAR-LINK, S, highlight list1 and press ENTER.
             b)  Press the multiply sign, x; then press 2ND, VAR-LINK, S, highlight list2 and press ENTER.   You
                   should now have list1*list2 displayed on the Home screen.
             c)  Press 2ND, STO, 2ND, VAR-LINK, S, highlight list3 and press ENTER..  You should now have
                   list1*list2→list3 pasted to the home screen.
             d)  Press ENTER and you will have the individual values stored in list list3 and displayed on the
                   home screen.  This is the products xp(x).
             e)  To get the sum of these values,  do this.
                      (1)  Press 2ND, MATH, 3, 6.  The expression sum( will be pasted to the Home screen. 
                      (2)  Press 2ND, VAR-LINK , l (L, not 1);  select list3 and  press ENTER. 
                      (3)  Close the parentheses and press ENTER
            You can obtain the variance and standard deviation by first solving for the variance using
             the formula:  Σx2 P(x) - 2 where 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)  Press 2ND, VAR-LINK, l (L, not1),  select list1 or whatever the x-values are in and press ENTER. 
              b)  Press ^, 2, x (multiply sign), 2ND, VAR-LINK, l (L, not1), highlight list2 and press ENTER.
              c)  Press STO, 2ND, VAR-LINK, cursor to list3 and press ENTER.  You will have list12*list2→list3
                   pasted to the Home screen.
              d)  Press ENTER and the individual values will be entered in  list3 and pasted to the home
                    screen.
              e)  To get the sum of these values do the following:
                      (1)  Press 2ND, MATH, 3, 6.  The expression sum( will be pasted to the home screen. 
                      (2)  Press 2ND, VAR-LINK, l (L, not1),  select list 3, close the parentheses  and press ENTER. 
              f)  Now subtract the value for 2 from the last value obtained and that will be the variance.
             g)  To obtain the standard deviation from the variance given in the 1-Var Stats, do the following:
                     (1)  From the home screen, press 2ND, √ , the multiply key; then enter the number for the variance,
                                close the parentheses, and press 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 use method 15 above.   Just for the  information of the reader,  the
            total expression for the variance using this method would the this:  sum(list12*list2) - (sum(list1 *list2))2 .

 III. 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=  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 APPS, S,  highlight the Stats/List icon and press ENTER.   If the tabulated lists do not appear, press
            ENTER again. 
       c)  Enter the data-point numbers ( the x-values)  in list1 and the corresponding values (y-
             values)  in list2.  (If your data is not in order you can sort in order by placing the cursor in the list
             that you want to sort and pressing F3, 2, 1.  The list where the cursor is located should be in the
             box opposite "List."  Move the cursor to "Sort Order" and select either ascending or descending
             as you choose.   Press ENTER to sort the list.  BE CAREFUL!  If your data in list2 is not in ascending order
             when correlated to list1, then your data in list1  and list2 will not be correlated correctly after sorting.
      d)  Press F2, ENTER to go to the Plot Setup screen.
      e)  Using the right cursor arrow, change the Plot Type to Scatter if it's not already there; then enter list1 in the
           box opposite "x" and list2 opposite "y." 
      f)  Press ENTER; then at the "Plot Setup" screen that appears, press F5 and the scatter points will be plotted.

2)   Plotting  x-y line Graph:
     
D
o that the same as the scatter plot in item 1 above except that when you select the "Plot Type,"  choose
      2: xyline rather than scatter. 

3)  Regression Analysis
     
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 APPS, S,  highlight the Stats/List icon and press ENTER.   If the tabulated lists do not appear, press
            ENTER again. 
      b)  Enter
  the numbers for the independent variable, x-values,  in list1 and the corresponding values in list2.
      c)   From the list screen, press F4, 3 and choose the regression equation of your choice, for example 2, for
            LinReg (ax+b).  Press the number for your selection.
      d)  On the dialog box that appears, enter list1 opposite "x" and list2 opposite "y."  Enter list1 by pressing
            2ND, VAR-LINK, l (L, not 1), highlight list1 and press ENTER.  Move the cursor down opposite "y" and
            repeat the steps to enter list2 there.
       e)  If you are planning to graph the equation,  the box opposite "Store RegEq to"
            should have  y1(x) in it.  If not, press the right arrow and then ENTER.  Enter 1 for Freq. if in is not already
            there  and press ENTER. 
       f)  The values for a, b, r and r2 will appear on a dialog box. 
            Graphing the Regression Equation:
       
a)  If you want to graph the regression equation, press
♦, Y=.  That will take you to the Y= screen.
      b)  Clear or deselect all of the entries except Y1 and check y1by pressing F4 if it's not already checked.
      c)  Press
♦, GRPH and the graph will be displayed.  You may need to adjust the window to see the graph.

4)  Plotting a graph with the scatter plot and the regression equation on the same axis.
     First you need to get your data in lists and do the regression graph as described above in item 3. 
     Make sure you have told the calculator where to store the regression equation in step e)  above.  Now, you want
     to put the  scatter plot on the screen with the graph. To do this:
    
Scatter Plot
      We will assume that your data are already in list1 and list2.
      d)  Press F2, ENTER to go to the Plot Setup screen.
       e)  Using the right cursor arrow, change the Plot Type to Scatter if it's not already there; then enter list1 in the
           box opposite "x" and list2 opposite "y." 
       f)  Press ENTER; then at the "Plot Setup" screen that appears, press F5 and the scatter points along with
           the graph of the regression equation will be displayed. 

 IV.  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.
  1.   Putting Data in Order:
          a)  Place the cursor in the list you want to sort and press F3, 2, 1. 
          b)  Move the cursor down opposite Sort Order and select either ascending or descending as you prefer by
                pressing the right cursor key if you need to change the entry.
          c)  Press ENTER and the list will be sorted. 

2.  Finding Mean (x-bar), ∑x, or ∑x2 , σ, Median, Q1, Q3 for Grouped or Ungrouped Data.
    For Ungrouped Data:
       
a)  Press APPS, S and select the Stats/List icon; then press ENTER.  This will take you to the tabulated list
                screen where you can enter your data
   
      b)  The list tables may be displayed, if not and your list tables are in the main folder, press ENTER.  If the lists
                are not in the main folder, select the correct folder and press ENTER.
          c ) 
Enter your data into list1 or whatever list you choose
          d)  Press F4, 1 for 1-Var Stats.
          e)  On the dialog box that appear, enter list1 in the box opposite "Lists," and enter 1 opposite the "Freq."
                box.
          f)  Press ENTER and various statistics will be displayed.
         
  NOTE:  You can also find these values for discrete random variable statistics by entering the values
                     of the variable in list, for example, and the probabilities corresponding data values in list2.
    For Grouped data:
        
a)  Press APPS, S and select the Stats/List icon; then press ENTER.  This will take you to the tabulated list
                screen where you can enter your data
   
      b)  The list tables may be displayed, if not and your list tables are in the main folder, press ENTER.  If the lists
                are not in the main folder, select the correct folder and press ENTER.
          c ) 
Enter your x-values  into list1 and the frequencies  in list2, or whatever lists you choose
          d)  Press F4, 1 for 1-Var Stats.
          e)  On the dialog box that appear, enter list1 in the box opposite "Lists," and enter list2 opposite the "Freq."
                box.
          f)  Press ENTER and various statistics will be displayed.

3.  Finding products such as xy, (x-y):
     a) Assume that your x-data is in list1 and your y-data is in list2. 
     b)  Press Home; then obtain the product by pressing 2ND, VAR-LINK, l (L, not1), scroll to list1 if not already
          there, enter x (multiply symbol), 2ND, VAR-LINK, l (L, not1)  ENTER.
     c)  Press ENTER and the result will be displayed on the home screen.
     d)  If you prefer to have the data stored in a list, list3 for example, before pressing ENTER in item c, press  STO,
            
2ND, VAR-LINK, l (L, not1), cursor to list3 and press ENTER.  
      
e)  Then press ENTER and the results will be displayed.
     f)  Obviously, x-y (x minus y) can be obtained by merely substituting the subtraction symbol for the
          multiplication symbol in step a) 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 list1.
     b)  Press 2ND, VAR-LINK, l (L, not 1), select the list you want and press ENTER.
     c)  Press ^, 2, ENTER.  The squared elements will be displayed.
     d)  If you want to store the squared data in a list, for example list3, then before pressing ENTER in
          item c above, press STO, 2ND, VAR-LINK, l (L, not 1), cursor to the list you want and press ENTER. 
          Press ENTER again and the data will be stored in list3.
     e)  If you want to multiply corresponding elements and square each product, your entries should result
          in the following:  (list1*list2)^2.  

5.  Find x-x (Sorry, I have no symbol for the mean, so I displaced the bar.) from the data in
     a list.
    
a)  Enter the data in list1. 
     b)  From the Home screen, press 2ND, VAR-LINK, l (L, not1), cursor to list1 and press ENTER; then press the
          minus sign not a negative sign.
     c)  Press 2ND, MATH, 6,4.   You should now have "list1-mean(" pasted to the home screen.
     d)  Press 2ND, VAR-LINK, l (L, not 1), highlight list1 and press  ENTER. 
     e)  Close the parentheses and press ENTER.  The result will be displayed on the home screen. 
     f)  If you want to store the results in a list, for example list3, then before ENTER in item "e" above, press
         STO, 2ND, VAR-LINK, l (L, not1), cursor to list3 and press ENTER.
     g)  If you did not have Exact/Approximate in the MODE set to Approximate, you will have some terrible looking
         fractions.  To avoid that, press MODE, scroll down to Exact/Approximate, press the right cursor arrow, and
         then press 3.

 6.  Finding (x-x )2  and Σ(x-x)2
     a)  Enter the data in list1. 
     b)  From the Home screen, press (, 2ND, VAR-LINK, l (L, not1), cursor to list1 and press ENTER; then press the
          minus sign not a negative sign.
     c)  Press 2ND, MATH, 6,4.   You should now have "list1-mean(" pasted to the home screen.
     d)  Press 2ND, VAR-LINK, l (L, not 1), highlight list1 and press  ENTER. 
     e)  Close the parentheses; then press ^, 2.  You should have the expression (list1-mean(list2))^2 displayed
          on the Home screen.
     f)  Press ENTER.  The result will be displayed on the home screen. 
     d)  If you want to store the results in a list, for example list3, then before ENTER in item "f" above, press
         STO, 2ND, VAR-LINK, l (L, not1), cursor to list3 and press ENTER.
     f)  If you did not have Exact/Approximate in the MODE set to Approximate, you could have some terrible looking
         fractions.  To avoid that, press MODE, scroll down to Exact/Approximate, press the right cursor arrow, and
         them press 3.
     g)  The expression
Σ(x-x)2 is evaluated in the 1-Var Stats.  To get that do the following:
               (1)  Press APPS, S and select the Stats/List icon; then press ENTER.  This will take you to the tabulated list
                      screen.  Enter your data in a list, for example, list1.
               (2)  Press F4, 1.  This will take you to the 1-Var Stats dialog box. 
               (3)  Opposite "List" enter the list number where your data is stored, for example list1.  You can either type in
                      the name or press 2ND, VAR-LINK, l (L, not 1), and press ENTER when the correct list number is
                     highlighted. 
               (4)  Type 1 in the box opposite Freq. and press ENTER.  If the statistic do not appear, press ENTER again.
                (5)  Cursor down and you will find Σ(x-x)2 listed. 

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. 
      b)  Press Home, (,  2ND, MATH, 6.  The expression "(sum(" should be entered on the home screen.
      c)  Press 5. 
      d)  Press 2ND, VAR-LINK, l (L,not 1), cursor to list1 (or whatever list you choose, and press ENTER.
      e)  Press ), ), ^,2 .  You now should have (sum(list2))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 by pressing F4, 1, but you can also
          find it by entering "sum list12 "

8.  Finding the Standard Deviation by hand using the above exercises:
    
The standard deviation computation formula is as follows:
     s = √(Σx -(Σx))/n)/(n-1) 
       So, one can use Σx2 and  (Σx)2 calculated above to calculate the sample standard deviation.

9.  Notice that you may also do several other operations by doing the "1-Var Stats" which is covered in another section
     of this document. 

V.  Permutations, combinations, factorials, random numbers:
 1. Finding Permutations.
    a)  Suppose we want the permutations (arrangements) of  8 things 3 at a time. Go to the home 
         screen.
    b) Press 2ND, MATH, 7, 2. You will have nPr( pasted to the Home screen.
    c)  Enter 8, 3. ), so that you have nPr(8, 3),  and press ENTER.  You will get 336.

 2. Finding Combinations:.
     a)  Suppose we want the combinations (groups) of  8 things 3 at a time.  Go to the home screen.
     b) Press 2ND, MATH, 7, 3. You will have nCr( pasted to the Home screen.
     c)  Enter 8, 3. ), so that you have nCr(8, 3),  and press ENTER.  You will get 56.     

 3. Finding Factorials.
     a)  Suppose we want 5 factorial (5!).  Go to the home screen.
     b)  Enter 5, ♦; then press the divide key.  You will have 5! pasted to the screen.
     c)  Press ENTER and the answer, 120, will be displayed.

 4.  Generating a random number between 1 and n:
     
a)  Suppose we want to generate a random number between 1 and 15.  Go to the Home Screen.
      b)  Press 2ND, MATH, 7, 4 and you will have rand( pasted to the home screen.
      c)  Enter 15, ) and press ENTER.  A random number between 1 and 15 will be displayed.

 5.  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 list1.  The proper syntax is randint(lower,
      upper, how many). 
That can be obtained as follows starting from the Stats/List.
      a)  Clear list1 by highlighting the title, list1, press Clear, ENTER.  Highlight the title again while
           you do the additional steps below.
      b)  Press F4, 4, 5.  The expression randInt( will be displayed at the bottom of the screen after list1=.
      c)  Enter 1, 50, 10, ), so that your screen displays randInt(1,50,10).
      e)  Press ENTER and the numbers generated will  will be stored in list list1.

6.  Normally distributed data set:
    
Suppose you want a set of 10 numbers from a data set with mean 50 and standard deviation 10.
     The proper syntax is randNorm(mean,std. dev., quantity)  That can be obtained as follows starting from the
      table of lists:
      a)  Clear list1 by placing the highlighting the title, list1, press Clear, ENTER.  Highlight the title again while
           you do the additional steps below.
      b)  Press F4, 4, 6.  The expression .randNorm( will be displayed at the bottom of the screen after list1=.
      c)  Enter 50, 10, 10, ), so that your screen displays .randNorm(50, 10,10).
      d)  Press ENTER and the numbers generated will  will be stored in list list1.

VI.  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 1 and a standard deviation of 0. 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
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.
       
a)  From the table of lists, press F5, 4 and  a dialog box for  "normalcdf" will appear.
        b)  Enter the number for the lower value, Upper Value, μ, and σ in that order. 
        c)  Press ENTER, and the value of the area between the two points 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 65 for Lower Value, 80 for Upper Value, 70 for mean, and 4.5 for σ.
          c)  Press ENTER ( you may need to press ENTER again) and 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 above problem, determine the probability that the value is less than 62.
           a)  Complete item a) in the general method above.
           b)   Enter numbers -1E9, for Lower Value, 62 for Upper Value, 70 for mean, and 4.5 for σ.  Notice that
               the "-" is a negative sign, not a minus sign.  Enter "E" by pressing the  EE button.
          c)  Press ENTER ( you may need to press ENTER again) and you'll get 0.03772 which is, of course,
               3.772 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 75 for Lower Value, 1E9 for Upper Value, 70 for mean, and 4.5 for σ.  Notice that
               the "-" is a negative sign, not a minus sign.  Enter "E" by pressing the  EE button.
          c)  Press ENTER ( you may need to press ENTER again) and you'll get  0.13326 which is, of course,
              13.326 per percent.

     4.   ShadeNorm(:  Displaying a graph of the area under the normal curve.
           General: 
This function draws the normal density function specified by
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
and σ defaults to a normal curve.  The following
          instructions, "a" through "c," are general instruction  to follow.

         
 a)  First turn off any Y= functions that may be active.  Do this by pressing ♦, F1 and either pressing
              F4 to disable each function or press clear to erase the function. 
           b)  Press 2ND, APPS to go to the list tables.  If this doesn't work, press APPS, highlight the Stats/List
                icon and press ENTER to go to the list tables.

           c)  Press F5, 1,1 and the Shade Normal dialog box will appear. 
               home screen. 
           d)  Enter your parameters, for example -1E9 for Lower Value, 62 for Lower Value, 70 for
, and 4.5
                     for
σ. 
           e)  Cursor down to Auto Scale and change it to YES by pressing the right cursor arrow and selecting YES.
            f)  Press ENTER, and the graph may be visible on the screen.  You may want to reset the WINDOW, but
                with the setting on Auto-scale, the graph usually looks satisfactory. 
              Ex 1:  Draw the graph of example 2 above.
               a)  From the Lists screen, press F5, 1,1
               b)  Enter your paramenters in the dialog box, for example -1E9 for Lower Value, 62 for Lower Value, 70 for
,
                         and 4.5  for
σ.
               c)  Cursor down to Auto Scale and change it to YES by pressing the right cursor arrow and selecting YES.
               d)  Press ENTER and a reasonable looking graph should appear on the screen.

       5.  invNorm(:  Inverse Probability Calculation: 
            Find the number x, in a normal distribution such that a number is less than x with a given 
            probability.  The default is the standard curve with mean=0 and standard deviation. is  1.
           Ex. 1:   In Ex. 1 immediately above, find the number x, such that a randomly selected number will be below
             that number with a 90% probability. 
            a)  Press F5, 2, 1 and the Inverse Normal dialog box will appear.
            b)  Enter numbers .9 for Area, 70 for
, and 4.5 for σ.
            c)  Press 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 Xo such
               that the probability of the given x-value is below Xo is .6523.  That is P(X<Xo) = .6523.
                 a)  Press F5, 2, 1 and the Inverse Normal dialog box will appear.
                 b)  Enter numbers .6523 for Area, 100 for
, and 20 for σ.
                 c)  Press ENTER, and perhaps ENTER again,  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 F5, 2, 1 and the Inverse Normal dialog box will appear.
                 b)  Enter numbers .9,  (1-.1)for Area, 70 for
, and 12 for σ.
                 c)  Press ENTER, and perhaps ENTER again,  and your answer will be  be 85.38 or 86 rounded off.

        6.  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 F5, 1, 1 and the Shade Normal dialog box will appear.
             b)  Enter numbers -1E9 for Lower Bound,  75.766 for Upper Bound, 70 for
, and 12 for σ.  Be sure
                  that Auto-Scale is set for YES.  Do that by pressing the right cursor arrow and selecting YES.
             c)  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 F5, 1, 1 and the Shade Normal dialog box will appear.
             b)  Enter numbers 40 for Lower Bound,  54 for Upper Bound, 46 for
, and 8.5 for σ.  Be sure
                  that Auto-scale is set for YES.  Do that by pressing the right cursor arrow and selecting YES.
             c)  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.

        7.   normalpdf(:  Probability Distribution Function using normalpdf( :
             General: 
This function is 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 F5, 3 and the Normal Pdf dialog box will appear.
                 b)  Enter numbers 50 "X value" ,  60 for
, and 12 for σ.
                 c)  Press ENTER and your answer should be .023493 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.  Set Ymin=0 and Ymax = .05
                 b)  Press ♦, F1 to go to the graphing screen.  Select position y1. 
                 c)  Press CATALOG, F3, n, select normpdf( and press ENTER.
                      the Y1= position.
                 c)  Enter data so that the entry after Y1= looks line this:  TIStat.normalpdf(X, 60,12).  Be sure to
                      close the parentheses.
                 d)  Press  ENTER to transfer the entry to y1; then press F2, ALPHA, a, to select ZoomFit. The curve should
                      appear on the screen.
                 e)  Press F3 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.

           8. ZInterval:  This 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 s = 4.61.  What is the 95%
                confidence level interval?
                   a)  From the lists screen press 2ND, F2, 1 and the ZInterval dialog box will appear.
                   b)  On the screen that appears, select "Stats" and press ENTER.
                   c)  Enter data opposite positions as follows:  σ: 4.61, x :15.58, n:90, and C-Level: .95.
                   d)  Press ENTER, and the interval (14.628, 16.532) will appear along with  values of other parameters.
              Ex. 2:  Suppose that you have a set of 35 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 list1, by pressing 2ND, VAR-LINK, l (L, not 1),  ENTER.
                        Enter you data into list1.  Just enter a data point and press either ENTER or the down arrow.  Leave
                        the cursor in in the list that your data is in.
                   b)  Press 2ND, F2, 1. 
                   c)  Select "Data" on the dialog box that appears and press ENTER.
                   d)  Next you need to know the sample standard deviation.  To enter that opposite σ, do this:  Press 2ND,
                        MATH, 6.  You should now have stdDev( in the box opposite σ. 
                   e)  Press 2ND, VAR-LINK , l (L, not1), ENTER.  You should now have stdDev(list1 pasted in the box
                        oposite σ.  Close the parentheses and move down to "List."
                    f)  Enter information as follows:  List: If the correct list is not entered, press 2ND, VAR-LINK, l (L, not 1),
                        move to the list you want and press ENTER.  Move down and enter Freq: 1, C-Level: .95.  Be sure
                        that the two list names are the same.
                   g)  Press ENTER.  The same type data will be displayed as in Ex. 1 above.

VII. Other Distributions and Calculations:

         1. TInterval:  If the sample size is <30, then the sample mean cannot be used for the population mean,  and
             the 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)  From the lists screen, press 2ND, F2 (for F7), 2.
                   b)  On the screen that appears, use the cursor to to select "Stats" and press ENTER.
                   c)  Enter data opposite positions as follows:  x :98.44, S x : .3   n:10, and C-Level: .95.
                   d)  Press ENTER, and, after a few seconds, the interval {98.23, 98.65}  will appear along with the values
                        for  "n" and the mean and a few other parameters..
              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 list1, by pressing the "Stat/List icon.  Now press
                       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 2ND, F2, 2  to bring up the "Choose Input Method"  screen.
                   c)  Use the cursor to set to to "Data" on the screen and press ENTER.
                   d)  On the TInterval screen that appears, enter information as follows:  List: Press 2ND, VAR-LINK, l (L, not 1).
                        If the cursor is not on list1, move it to list1 and press ENTER.  This will place list1 in the List box of the TIntrval
                       screen.  Now enter 1 opposite Freq: 1 and .95 opposite C-Level.
                   e)  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  where 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 F5, 6,  5, to bring up the tcdf( dialog box.
                 b)  Enter data in the boxes as follows:  Lower Value: 1.4, Upper Value 1E9, Deg of Freedom, df: 19.
                 c)  Press ENTER and your answer should be .0888.

           3.  invT: Finding a t-value Given α and df:
               
                 This calculator has an invT, so do the following:
                    (1)  Press F5, 2, 2, and the Inverse t dialog box will appear.
                    (2)  Enter α or 1-α, depending on whether you have a left or right tail; then enter the degrees of freedom, df.
                    (3)  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 F5, 8 to bring up the Chi-Square Cdf dialog box. 
                  b)  Enter data so that your display is as follows:  Lower Value:  24, Upper Value: 1E9, df: 19.
                  c)  Press ENTER and your answer should be .1961.

VIII.  Hypothesis Testing:
      1.  Testing for Mean  and z Distribution with Data:
           a)  Enter the data into list1 or whatever list you choose.
           b)  Press 2ND, F1, 1, ENTER and move the cursor over to TESTS.
           c)  When the "Input Method" dialog box appears, select "Data" and press ENTER.
           e)  Opposite
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,
                MATH, 6, and stdDev(, will be displayed opposite σ.  Now, enter your
                list number where the data is stored by pressing 2ND, VAR-LINK, l, (L, not1), select the appropriate list
               and press ENTER.  
           g)  Opposite "List," enter the list where you data is stored with the keystrokes mentioned above 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.

       2.  Testing for Mean and z Distribution with Statistics: 
           a)  Select the Stats/Lists icon and press ENTER.
           b)  Press 2ND, F1, 1  or ENTER for Z-Test.
           c)  Select "Stats" on the dialog box that appears and press ENTER.
           d)  Opposite
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.     

      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 F5, 2, 1 for invNorm(. 
           b)  Opposite "Area," enter the value for α for a left-tailed or 1-α for a right-tailed.  Clear the values for
                     and σ or enter 0 opposite and 1 opposite σ.
            c)  Press ENTER and the z-value 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, F1, 1, and invNorm( will be pasted to the home screen.
             b)  Enter the value for area, e.g., .05, : 5.25, σ: .6.  For a left tail, enter the value
                     for α and for a right tail enter 1-α..
             c) Press ENTER and the inverse will be displayed.  

       5.  Testing for Mean  and t Distribution with Data:
           a)  Enter the data into list1 or whatever list you choose.
           b)  Press 2ND, F1, 2 for T-Test. 
           c)  On the dialog box that appears, select "Data" and press ENTER.
           e)  Opposite
o, enter the mean for the null hypothesis.
           f)  Enter list1 or whatever list your data is in and opposite Freq. enter 1.
           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.  Refer to the procedures directly above to find these values with a
               TI-89 Titanium.

       6.  Testing for Mean and T Distribution with Statistics: 
        
           a)  Press 2ND, F1, 2 for T-Test.
           b)  On the dialog box that appear, select "Stats" and press ENTER.
           c)  Opposite
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, MATH, 6, 4 and Mean( will be pasted to the box opposite mean.
           f)  Press 2ND, VAR-LINK, l (L, not1), select the appropriate list  and press ENTER. Close the parentheses.
          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.  You can find the z-value or the critical x-value using the procedures
                     above in this section.

X.  Statistics of two Populations:   
     
1.  Confidence Interval for Two Dependent Populations (Data):
 
 Enter the data from population 1 into "list1" and the data from population 2 into List2.  Do this as follows:
     a)  Press APPS, select the "Stats/Lists" icon and press ENTER. 
     b)  Enter  the data in the displayed lists. 
   Now, store the paired differences in list list3 as follows:
     c)  From the Home  screen, press 2nd, VAR-Link, l (L not 1), select list1 and press ENTER.
    d)  Press the minus sign; then 2nd, VAR-Link, l (L not 1), select list2 and press ENTER.
    e)  Press STO, 2nd, VAR-Link, l (L not 1), select list3 and press ENTER. You should now have
          list1-list2 → list3 on the Home screen.
     f)  Press ENTER and the pared differences will be stored in list3.
  Now, find the confidence level as follows:
    g)  Press 2ND, F2, 2 TInterval.
    h)   On the screen that appears, press the right cursor arrow and select  "Data" and press ENTER, ENTER. 
    i)   In the dialog box that appears, enter list3 in the box, either by using the VAR-LINK method or by typing
           list3 by hand.
    j)  Enter  1 opposite "Freq," and the confidence level you want, for example .95, opposite "C-Level."
    j)  Press ENTER and perhaps ENTER again and 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.
      Find the confidence level from the list screen as follows:
        a)  Press 2ND, F2, 2 TInterval.
        b)   On the  that appears, press the right cursor arrow and select  "Stats" and press ENTER, ENTER. 
         c)   In the dialog box that appears, enter the sample mean, the sample standard deviation, the number of
               data points, and the confidence level you desire.  
         d)  Press ENTER and perhaps ENTER again and the confidence interval and other statistics will be
             displayed.   

 3.  Confidence Interval for Two Independent Populations (Stats):
 Find the confidence level from the list screen as follows:
        a)  Press 2ND, F2, 4 for "2-SmplTInt."
        b)   On the  that appears, press the right cursor arrow and select  "Stats" and press ENTER and ENTER
               again if a new dialog box does not appear.
        c)   On  the dialog box that appears, enter the sample mean, the sample standard deviation, the number of
               data points,  for both samples.  In the last space, enter  the confidence level you desire.  
       d)  Press ENTER and perhaps ENTER again and the confidence interval and other statistics will be
             displayed.   
     
 

4.  Confidence Interval for Two Independent Populations (Data):
    
First, we must enter  the data from population lists.  Do this as follows:
     a)  Press APPS, select the "Stats/Lists" icon and press ENTER, STAT, ENTER.
     b)  Enter  the data in the displayed lists, for example list1 and list2.. 
 Now, find the confidence level as follows:
    c)  Press 2ND, F2, 4 for "2-SmplTInt."
    d)   On the screen that appears, press the right cursor arrow and select  "Data" and press ENTER, ENTER. 
    e)   In the dialog box that appears, enter list1 in the box after "List 1," list2 after "List 2;" enter 1 in both the
          "Freq 1" and "Freq 2" boxes.  You can enter the list numbers either by using the VAR-LINK lists or by
          typing the lists in by hand.
    f)  Enter the confidence level you want, for example .95, opposite "C-Level."
   h)   Highlight "No" opposite "Pooled" if  there are no assumptions about the variations.
    i)   Press ENTER and if the screen does not change, press ENTER again. 

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