 
Brief User Guide for TI84 Statistics
Contents: This document
covers single and twovariable statistics, scatter plot, regression analysis,
normalpdf and
normalcdf,
Student's Distribution, geometric mean, and much more.
Date Last Revised: 5/9/15
INDEX:
For TI83 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.
SingleVariable 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. TwoVariable Statistics – scatter plot, regression analysis,
finding r, r^{2}, 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 tvalue given α
and df, Chisquared Distribution, binomialpdf, binomialcdf.
VII. Hypothesis testing  mean and ztest (data), mean and ztest
(statistics), mean and ttest (data),
mean and ttest (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  oneway ANOVA, ChiSquare test for
independence, X^{2 }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
Ʃx^{2} , (Ʃx)^{2
, etc using lists, }
XI. SingleVariable Statistics  Histogram by hand
Doing a Discrete Probability
Distribution by Hand
XII. Two Variable Statistics  finding r, r^{2}, a, and b in correlation
using a computation formula,
Finding the Values a and b for the BestFit 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 L_{1} ; 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}, L_{1} (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. SingleVariable
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 L_{1}, 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 L_{1}, 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 L_{1} (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 Xmin and press ENTER. A new value for Xmin
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 L_{1} is not already entered, enter it by pressing
2ND, L_{1}. 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}, L_{1} (the 1 key) to
display min(L_{1 }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 (4320)/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 xmin, xmax, and "n"
on a sheet of paper in tabular form.
b) Add onehalf the class width to each xmin value and record those
values. Store these values in a list, for
example
L_{2} if you have your histogram data in L_{1}.
Store the corresponding values of "n" in L_{3}.
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 L_{2 }opposite
Xlist and L_{3} opposite Ylist.
e) Press ZOOM, 9 and the graph will appear on the screen.
NOTE: Some teachers or texts prefer returntozero 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 L_{2}. To do that place the
cursor at the first item in L_{2}, press INS and replace the zero that
appears with the first midpoint you calculated. Go to the bottom
of the L_{2} list and enter the second value you
calculated.
B. Now you want to enter zero in L_{3} opposite each of these
new midpoints. Place the cursor at the top of L_{3} 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 L_{2}.
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, L_{4 }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, L_{1}. 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(L_{3})
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 L_{4 }opposite
Xlist and L_{5} opposite Ylist.
NOTE: If you did a returntozero graph for the frequency
polygon, go to the list and delete the last
midpoint and zero in L_{4 }and L_{5 }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 xvalues and frequencies, do the
steps listed for each type graph.
Relative Frequency:
Assume that we want to store the relative frequencies in list L_{3},
the frequencies are in L_{2,}and the xvalues are in L_{1} .
a) First place the cursor to highlight the list title, L_{3}.
Press 2ND, L_{2}, ÷, 2nd, LIST, move the cursor to
MATH and
press 5. You
should now have L_{2}/sum( displayed on the bottom of
the list screen.
b) Press 2ND, L_{2}, ), ENTER and the relative frequencies
will be stored in list L_{3}.
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 L_{1 }opposite
Xlist and L_{3 }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 L_{4 }and that the relative frequency
is still stored in L_{3} from the above relative
frequency operation above, and that the xvalues are in L_{1}.
a) First place the cursor to highlight the list title, L_{4}.
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, L_{3}, ), ENTER. The cumulative relative
frequencies will now be stored in L_{4}.
c)
Press 2nd, STAT PLOT, ENTER. If "On" is not highlighted, select it and press
ENTER.
d) Highlight the second icon, and enter L_{1 }opposite
Xlist and L_{4 }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 L_{1} and the
corresponding frequencies into L_{2} . You
can enter the arithmetic operations in the list if you prefer, e.g.
(upperlower)/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 L_{1 }opposite Xlist and L_{2 }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, L_{3 }if
you have data in the first two lists.
b) If you have the frequency in L_{2 }, place the cursor
over the list title, L_{4}, 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(L_{2}) 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 L_{3 }opposite Xlist
and L_{4 }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 L_{5} and
the upper limits on the classes are in L_{3} .
a) First place the cursor to highlight the list title, L_{5}.
Press 2ND, L_{3}, ÷, 2nd, LIST, move the cursor to
MATH and
press 5. You should now have L_{3}/sum( displayed on the bottom of
the list screen.
b) Press 2ND, L_{3}, ), ENTER.
c)
Press 2nd, STAT PLOT, ENTER. If "On" is not highlighted, select it and press
ENTER.
d) Highlight the second icon, and enter L_{3 }opposite
Xlist and L_{5 }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 L_{6 }and that the relative frequency
is still stored in L_{5} from the above relative
frequency operation above, and that the class upper boundaries are in L_{3}.
a) First place the cursor to highlight the list title, L_{6}.
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, L_{5}, ), ENTER. The cumulative relative
frequencies will now be stored in L_{6}.
c)
Press 2nd, STAT PLOT, ENTER. If "On" is not highlighted, select it and press
ENTER.
d) Highlight the second icon, and enter L_{3 }opposite
Xlist and L_{6 }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 L_{1} 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 L_{3} 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, L_{2} , ), ÷ . You now should have cumSum(L_{2})/
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, L_{2}, ). You now should have cumSum(L_{2})/Sum(L_{2})
at the bottom of the list screen.
g) Press x (the multiply symbol), 100. You now should have
cumSum(L_{2})/Sum(L_{2}) *100 at the bottom
of the list screen.
h) Press ENTER and the data will be stored in L_{3} .
Plotting the Data:
i) Press 2nd, STAT PLOT, ENTER
j) Select the second icon and enter L_{1} opposite Xlist and
L_{3} 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 xvalues 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 (MandM) 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 MandM.
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
boxandwhisker 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 L_{1}, 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 boxandwhisker 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/xbar, 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 L_{1}, 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, L_{1}, ), and then press the divide symbol.
e) Press 2nd, LIST, move the cursor to MATH, and press 3 for mean.
f) Press 2nd, L_{1} , close the parentheses and then.
You should now have StdDev(L_{1)}/mean(L_{1}). Press ENTER to
display the
CV as the last number in L_{1}. 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 L_{1} as described under Entering Data
immediately above.
d) Press STAT, move the cursor to CALC, and press ENTER. The
expression “1Var Stats”
should be pasted to the home screen along with the following list:
List:
FreqList:
Calculate
Enter the list, for example L_{1}, 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 L_{1}; then enter the
frequencies in L_{2}.
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 reenter 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, x^{2} , 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.
Classes 
Class
Midpoint x (L_{1}) 
Freq. (f) (L_{2)} 
3545 
40 
2 
4555 
50 
2 
5565 
60 
7 
6575 
70 
13 
7585 
80 
11 
8595 
90 
11 
95105 
100 
4 
c) Enter the class
midpoints in list L_{1}. You can either do
the midpoints by hand or calculate
and store them in list L_{1} as follows:
(1) Store the lower boundaries in list L_{1} and the upper
boundaries in L_{2}.
(2) Place the cursor on the title of L_{1}; then press (,
2ND, L_{1}, + 2ND, L_{2},), ÷,
2. You should have (L_{1} + L_{2})/2_{ }at
the bottom left of the tables. Press
ENTER and the midpoints will be stored in L_{1}.
d) Clear list L_{2} if necessary, and enter the
frequencies in L_{2} 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 “1Var
Stats” should be pasted to the home screen along with the following
list:
1Var 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 x^{2} 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 L_{1} and
the weights in L_{2}.
b) Press STAT, move the cursor to CALC, and press ENTER to paste
"1Var 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 xbar (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.
Age 
514 
1524 
2534 
3544 
4554 
Midpoint 
9.5 
19.5 
29.5 
39.5 
49.5 
Freq (People) 
750 
2005 
1950 
195 
100 
a) To
find the median class, enter the midpoints in L_{1} and the frequencies
in L_{2 }.
b) Press STAT, move the cursor to highlight CALC and press ENTER.
The following will be displayed on the home screen.
1Var Stats
List:L_{1}
FreqList:L_{2}
Calculate
c
If the list numbers are in other lists, you must enter the correct lists. For example, ^{ }
press 2^{ND}, L_{2}, comma, 2^{ND}, L_{3} 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 1524 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(5000/2750)/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.
FUND OR
BENCHMARK 
YEARLY RETUNS 
S& P 500 
10.88, 4.91,
15.79, 5.49, 37.00, 26.46, 15.06, 2.11, 16.00, 32.39 
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 oneyear
periods. So, we must use the geometric mean.
a) Press STAT, ENTER and enter the returns data in list L_{1} or whatever
list is convenient.
b) Place the cursor on the list name of list L_{2} or whatever list
you choose, and enter 2nd, L_{1}/100 +1 and press ENTER.
Now, we're going to use the formula (products of L_{2})^{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, L_{2} so that you have prod(L_{2 }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, L_{1}. The entries, randint(1,50,10)>L_{1},
will appear on the screen.
d) Press ENTER and the numbers generated will appear on the screen and
will be stored in list L_{1}.
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/}(_{N}C_{j})
Filling in the formula as follows:
P(k=5) = _{10}C_{5}*_{15}C_{7}/(_{24}C_{12})
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 setup should be as
follows:
_{10}C_{5}*_{15}C_{7}/(_{24}C_{12})
(Using p(k=5)
d) Press ENTER to get .59996....
IV.
Twovariable 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 datapoint numbers ( the xvalues) 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], L_{1}, 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 L_{1} in the Xlist, and L_{2} 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 xy 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 scatterdiagram icon.
3) Regression
Analysis:
Assume
that you have the following information on the heights and weights on a group of
young
women:

1 
2 
3 
4 
5 
6 
7 
8 
Height x 
65 
65 
62 
67 
69 
65 
61 
67 
Weight y 
105 
125 
110 
120 
140 
135 
95 
130 
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, xvalues, 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: L_{1}
Ylist: L_{2 }
FreqList_{ }
Store RegEq:
Calculate
d) If you want to graph the equation of the bestfit line, ship to item
“e” below. If you don't want to graph,
continue with these instructions. If you have your data in the L_{1} and
L_{2} 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, r^{2}, and r will
be displayed. Note that if r and r^{2} 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…,
r^{2} =.63366…, and r=.7979…
e) Now, you
want to store the equation as a Yvariable, say, Y1, do it this way: Press
[VARS], move the cursor to
YVARS, [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
r^{2 }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 Yvariable, 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 bestfit graph
will appear on the screen.
e) You can press [TRACE] to display the xy 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 xdistance (xmaxxmin) a multiple or submultiple 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

1 
2 
3 
4 
5 
6 
x 
9.9 
11.4 
8.1 
14.7 
8.5 
12.6 
y 
37.1 
43 
33.4 
47.1 
26.5 
40.2 
a) If the
data are not already in the lists, press STAT, ENTER and enter the xvalues in
list L_{1} and the
yvalues in list L_{2}.
b) Press STAT, move the cursor to TESTS, and scroll down to LinRegTTest.
Press ENTER and the following screen will be displayed.
LinRegTest
Xlist: L_{1}
Ylist: L_{2 }
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 xvalues 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
zvalues 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 onefiftieth 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 zvlaue
for a particular confidence level:
Suppose you want the zvalue 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 lefttailed or 1α for a righttailed. 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 zvalue 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 xvalue 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 yaxis 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 xvalues.
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 xvalue 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
CLevel: .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: L_{1}
Freq: 1
CLevel: .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, L_{1 }, or
whatever list you have your data in.
f) Enter additional information as follows: Freq: 1,
CLevel: .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 CLevel: .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, L_{1},
Freq: 1, CLevel: .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> 1.4df = 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 tvalue 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 1α, 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 Chisquared Distribution: The χ^{2} Distribution
is implemented similar to the Student's t
Distribution.
Ex. 1: Assume that you want to find P(χ^{2} > 24df=20)
the same as in the above Student's t Distribution.
a) Press 2nd, DISTR, 8 to paste χ^{2}cdf( to the home
screen.
b) Enter data so that your display is as follows:
χ^{2}cdf
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 L_{1}.
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, L_{1} 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 L_{1}. 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 ZTest.
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, xbar, and n.
f) Select the proper condition for the alternative hypothesis.
g) Move the cursor to Calculate and press ENTER. The zvalue,
pvalue and some other statistics will
be displayed.
2. Testing Mean of z Distribution with Data:
a) Enter the data into L_{1 }or
whatever list you choose.
b) Press STAT and move the cursor over to TESTS.
c) Press 1 or ENTER for ZTest.
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 L_{1} 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 zvalue or
critical value, see those procedures below.
3) Finding a zvlaue
for a particular confidence level:
Suppose you want the zvalue 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 lefttailed or 1α for a righttailed. 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 zvalue 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 1α
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 L_{1 }or
whatever list you choose.
b) Press STAT and move the cursor over to TESTS.
c) Press 2 for TTest.
The following screen will be displayed:
TTest
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 L_{1} 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 pvalue test, read the
pvalue and compare it with α
or α1 as appropriate.
j) If you are working a problem using the tvalue 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
tvalue 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 TTest. The following screen will be displayed:
TTest
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, xbar, and n. If you don't know
xbar 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 L_{1 }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 pvalue test, read the
pvalue and compare it with α
or α1 as appropriate.
k) If you are working a problem using the tvalue 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 TI83 Plus or a TI84 Silver Edition. See "invT:
Finding a tvalue 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 L_{1} and the data from population 2
into L_{2}. 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, L_{3 }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 L_{3}.
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: L_{1}
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 CLevel, 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:
CLevel:
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 CLevel, 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 "CLevel."
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 L_{1} and the data from population 2
into L_{2}. 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, L_{1} and opposite "List2,"
press 2nd, L_{2}.
e) Set the confidence level you choose opposite "CLevel."
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:
x 
10 
11.5 
8 
14.5 
8.4 
12.7 
y 
37 
43.2 
33 
46 
25.5 
41.2 
Let's calculate the standard
error of estimate, "s" on the TI84.
a) Press STAT, ENTER and enter the
xvalues in list L1 and the yvalues 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.
Oneway 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:
Group 1 
Group 2 
Group 3 
Group 4 
x_{1} 
x_{2} 
x_{3} 
x_{4} 
9 
2 
3 
5 
8 
4 
7 
3 
9 
3 
9 
7 
8 
5 
8 
7 
7 

5 
6 


6 

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 L_{1} through
L_{4}. 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 TI84.
Press ENTER and ANOVA( will be displayed on the home
screen.
c) Inter list numbers so that you have the following display: ANOVA(L_{1},
L_{2}, L_{3}, L_{4}. 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. ChiSquare 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.
Consumer Product 
Group #1 
Group #2 
Row Totals 
Brand #1 
1
O=33: E=36 
2
O=57: E=54 
90 
Brand #2 
3
O=30: E=20 
4
O=20 
50 
Brand #3 
5
O=5: E=16 
6
O=35 
40 
Brand #4 
7
O=12 : E=8 
8
O=8 :E=12 
20 
Column Totals 
80 
120 
200 
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 Χ^{2} 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. Χ^{2} =24.64 and
p=1.79467 will be displayed.
h) Since P is smaller than the αvalue 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 = (41)(21) and consulting a
table, we would have found Χ^{2} =11.34. Since the test
statistic is larger, we would reach the same conclusion as above.
3. Х^{2} Goodness
of Fit:
The
TI83 Plus does not have a GOF function, so I will first give the procedure for
the TI84; then I will give a procedure for the
TI83 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.
Frequency 
Red 
Blue 
Green 
Grape 
Lime 
Observed 
30 
28 
20 
15 
12 
Expected 
21 
21 
21 
21 
21 
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 L_{1} and the expected
values in L_{2}.
b) Press STAT, move the cursor to TESTS, and cursor down to D: Х^{2} GOF
– TEST and press ENTER.
Alternately, you can press ALPHA, D to activate that procedure.
c) On the screen that appears, make sure that L_{1} is opposite
Observed and L_{2} 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 L_{1}*L_{2} with
the L_{3} title highlighted will
do the same thing at the list screen as L_{1}*L_{2}→L_{3} 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, L_{1} (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, L_{1} and L2_{,
}for example, use SortA(L_{1},L_{2}).
2. Finding Mean
(xbar), ∑x, or ∑x^{2} , σ, Median, Q_{1}, Q_{3} 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. "1Var 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 L_{1}, for example, and the
corresponding data values in L_{2}.
For Grouped data:
a) Find the midpoints of each group and enter those values in L_{1}.
You can have the calculator
do the arithmetic by entering the two values divided by 2. Finally, enter the
corresponding frequencies in
L_{2}. 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 "1Var Stats" screen will be pasted to the home screen. _{
}c) Move the cursor opposite List and press 2nd, L_{1}; then
move the cursor opposite FreqList and press 2nd, L_{2}. 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 ∑x^{2 }f is listed as ∑x^{2} .
3. Finding results such
as xy or (xy):
a) Assume that your xdata is in L_{1 }and your ydata is in L_{2}.
Then obtain the product by pressing
2nd, L_{1}; x (multiply symbol), 2nd, L_{2}, ENTER.
b) If you want the data stored in a list, L_{3 }for example, first
press STATS, ENTER and highlight the list name L_{3}.
Now, press 2nd, L_{1}, x (Multiply symbol), 2nd, L_{2. }Then
press ENTER.
c) Obviously, xy 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 L_{1}.
b) Press 2nd, L_{1}; then the x^{2} symbol, ENTER. The
squared elements will be displayed.
c) If you want to store the squared data in a list, for example L_{3},
first press STATS, ENTER and highlight the list name L_{3}.
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: (L_{1 }* L_{2})^{2} .
5. Find xx¯ (Sorry, I
have no symbol for the mean, so I displaced the bar.) from the data in
list L_{1}
and store the results in L_{2.}
a) Enter 2nd, L_{1}, . Note that" " is a minus sign not a
negative sign.
b) Cursor to MATH and press 3. You should now have "L_{1}mean(" pasted to the
home screen.
c) Press 2nd, L_{1}, ENTER. The result will be displayed on the
home screen.
d) If you want to store the results in a list, for example L_{3},
highlight the list name where you
want the data stored; then enter the operation as described above.
Finally, press ENTER.
6. Finding (xx¯ )^{2 }
a) Move the cursor to the title for L_{2 }and press (, 2nd, L_{1},
. Note that the ““ is a minus sign.
b) Press 2^{nd}, STAT, move the cursor to MATH and press 3. You should
now have "(L_{1}mean(" pasted to the home screen.
^{ }c) Press 2nd, L_{1},),),x^{2} . The
expression (L_{1}mean(L_{1}))^{2} should now be
displayed on the screen.
Press ENTER and the results will be displayed in L_{2}.
Note that this can also be done from the home screen if desired.
7. Finding (Σx)^{2} and
Σx^{2}
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, L_{1} or whatever list your data is stored in.
e) Press ), ), x^{2} . You now should have (sum(L_{1}))^{2} on
your home screen.
f) Press ENTER and the results will be displayed on the screen.
g) Σx^{2} can be found by using the "1Var Stats" function
under STATS, CALC, but you can also
find it by entering "sum L_{1}^{2} "
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 L_{1}
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(L_{1}
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 L_{1} (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 Xmin and press ENTER. A new value
for Xmin 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 xmin
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 Q_{1}, Median, and Q_{3}
when doing a boxandwhisker 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 "1Var Stats" screen will be
displayed on the home screen.
b) If your data is in list L_{1} move the cursor opposite
List and press 2^{nd}, L_{1}; then move the
cursor to Calculate and press ENTER.
c) The values for Xmin, Xmax, Q_{1}, Q_{3}, 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 L_{3},
do the following: (The xvalues should
should be stored in L_{1} and the p(x) values in L_{2}.)
a) From the list screen, highlight the title of L_{3} and
press 2ND, L_{1}, x, 2ND, L_{2}.. You will now have L_{1}*L_{2
} 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 L_{3}. 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, L_{3} , ).You will have sum(L_{3})
at the bottom of the list screen.
(3) Press ENTER and the sum of those values will be
displayed as the last item of L_{3}.
You can obtain the variance and standard deviation by first solving
for the variance using the formula: Σx^{2} P(x)
 µ^{2}
where µ is the mean obtained as above. To obtain the individual
values of the first term, x^{2} P(x), and store them in list L_{4},
do the following:
a) From the list screen, place the cursor on the title for list L_{4} ,
press 2ND, L_{1}, x^{2}, ,x, 2ND, L_{2}. You will
have L_{1}^{2}*L_{2} 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 L_{4}, 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(L_{4})
at the bottom of the list screen.
(3) Press ENTER and the sum of those values in L_{4} will
be displayed as the last entry in L_{4.}.
d) Now we want to subtract the value for µ^{2} 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 L_{4}, Σx^{2} 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 L_{3}(7), move the cursor down
one
space to L_{3(}8)and press 2ND, √, 2nd, L_{3},(,7,)
and press ENTER. The standard deviation will be displayed in L_{3}(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 1Var 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
TI84 for doing much
of the arithmetic required and obtaining numbers to plug into the formulas.
Suppose that students did situps according the table shown below.
Student 
Situps (x) in (L_{1}) 
x^{2} in ^{(}L_{2}) 
1 
22 
484 
2 
27 
729 
3 
15 
225 
4 
35 
1225 
5 
30 
900 
6 
52 
2704 
7 
35 
1225 



n=7 
Σx=216 
Σx²=7492 
The variance
computation formula is as follows: s^{2} = [(Σx²
(Σx)²)/n)]/(n1), where s^{2} is the variance .
So, we will need x^{2} , ∑x^{2} 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 “1Var Stats” screen
should be
pasted to the home screen. Enter
information so that the appearance is as follows:
1VarStats
List: L_{1}
FreqList:
Calculate
Move the cursor to Calculate and press ENTER. The numbers listed
below will be displayed.
c) Copy n=7, ∑x = 216, and ∑x^{2} =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 x^{2} column. Place the cursor on the title L_{2},
press 2^{nd}, L_{1}, x^{2}, ENTER. The squares of the
numbers in L_{1} will be displayed in L_{2}. You can
enter into your table the numbers that you found for n, ∑x, and
∑x² from the 1Var 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 (7492216^{2}/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
xvalues 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.
Classes 
Class
Midpoint x (L_{1}) 
Freq. (f) (L_{2)} 
xf
(L_{3}) 
_{
x}2_{f
}(L_{4}) 
3545 
40 
2 
80 
3200 
4555 
50 
2 
100 
5000 
5565 
60 
7 
420 
25200 
6575 
70 
13 
910 
63700 
7585 
80 
11 
880 
70400 
68595 
90 
11 
990 
89100 
95105 
100 
4 
400 
40000 


n=Σf=50 
∑x=Σxf=3780 
∑x^{2} =^{ } Σx²f=296600 
The formula for
the grouped data variance is this:
s^{2} =( Σx^{2}
(Σxf)^{2} /Σf)/(Σfa)
a) You can either do the midpoints by hand and store them in L_{1} or
calculate and store them in list L_{1} as follows:
(1) Store the lower boundaries in list L_{1} and the upper
boundaries in L_{2}.
(2) Place the cursor on the title of L_{1}; then press (,
2ND, L_{1}, + 2ND, L_{2},), ÷,
2 . You should have (L_{1} + L_{2})/2_{ }at
the bottom left of the tables. Press
ENTER and the midpoints will be stored in L_{1}.
Now let’s calculate the required numbers.
b) Press STAT, move the cursor to CALC, and press ENTER. The
expression “1Var 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 L_{1}; then
enter the frequencies in L_{2}.
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,
∑x^{2}=∑x^{2}f=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 ∑x^{2}f is
listed as ∑x^{2}.
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 L_{3} by
placing the cursor over the title for L_{3 }pressing 2ND, L_{1},
*, 2ND, L_{2}._{.}
You should have L_{1*}L_{3} at_{ }the bottom left
of the tables. Press ENTER and the products will be stored
in list L_{3.}
e) Calculate x^{2}f by placing the cursor on the title for L_{4}
and pressing 2ND, L_{1, }x^{2} , * , 2ND, L_{2}.
You should now have L_{1}^{2}_{ }*L_{2} 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 (2966003780²/50)/(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 BestFit Equation^{ }Using a Computation Formula:
We
will use calculator functions to reduce the arithmetic necessary for these
formulas. First we will
use "2Var 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:

1 
2 
3 
4 
Height x 
65 
65 
62 
67 
Weight y 
105 
125 
110 
120 

5 
6 
7 
8 
x(Cont'd) 
69 
65 
61 
67 
y(Cont'd) 
140 
135 
95 
130 
The
formula for “b” is this: (nΣxy –ΣxΣy)/(nΣx^{2} (Σx)^{2}). So,
you will need to record the values
for xbar, ybar, Σx, Σy, ΣxΣy, Σx^{2}, Σy^{2}, and n..
You can get all of these by using the 2Var Stats function.
Use that as follows:
a) With the data in lists L_{1} and L_{2} press STAT,
move the cursor to CALC, and press 2. The
expression 2Var 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, Σx^{2}=33979,
n=8, Σy=960, ybar=120, Σy^{2}=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
theTI84, 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)/(nΣx^{2} (Σx)^{2}).
Now record these numbers on your worksheet or in the calculator if
you're going to calculate the values.:
=(8*62750521*960)/(8*33979521^{2})
=4.7058…..
e) Now, record this formula on your worksheet:
a= ybar –b(xbar)
Now just enter these numbers on your worksheet or enter them in your
calculator if you're going to calculate the value.
=1204.7058 *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 ∑x^{2} into matrix element 2,2. Display the
expression once again and edit it using the listings under VARS, Statistics to
get ∑x^{2}à[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 r^{2 }Using a Computation Formula:
We
will use calculator functions to reduce the arithmetic necessary for these
formulas. First we will
use "2Var 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:

1 
2 
3 
4 
5 
6 
7 
8 
Height x 
65 
65 
62 
67 
69 
65 
61 
67 
Weight y 
105 
125 
110 
120 
140 
135 
95 
130 
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, xvalues, in L1 and
the corresponding
yvalues in L2.
NOTE: The formula for “r” is this: (nΣxy –ΣxΣy)/[(√nΣx^{2}
(Σx)^{2})(√nΣy^{2} (Σy)^{2})]. So, you will
need Σx, Σy, ΣxΣy, Σx^{2}, Σy^{2,}, and n. You can
get all of these by using the 2Var Stats
function. Use that as follows:
b) With the data in lists L_{1} and L_{2} press STAT,
move the cursor to CALC, and press 2. The
2Var Stats
screen wills be displayed on the screen as follows:
2VarStats
Xlist: L_{1} _{
} 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, Σx^{2}=33979, n=8, Σy=960,
Σy^{2}=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
TI84, 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: L_{1} _{
} 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)/[(√(nΣx^{2} (Σx)^{2})(√(nΣy^{2}
(Σy)^{2})]
= (8*62750521*960)/(√(8*33979521^{2})(√(8*116900960^{2}))
=.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*62750521*960, STO, ALPHA, N.
2) Calculate the denominator and store it in two separate variables M
and D. In this manner
√(8*33979521^{2} ) , STO, ALPHA, M; then √(8*116900960^{2}),
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(L_{1})
:min(L_{1})>S
:dim(L_{1})>Q
:max(L_{1})>M
:int(M/W)+1>dim(L_{2})
:Input "CLS WDTH ",W
:0>T:1>X:W>F:0>C
:ClrHome
:Lbl 1
:While L_{1}(X)≥S
and L_{1}≤F
:T+1>T
:X+1>X
:If X>Q
:Then
:T>L_{2}(C+1)
:Goto 2
:End
:End
:C+1>C
:T>L_{2}(C)
:0>T
:S+W>S
:F+W>F
:Goto 1
:Lbl 2
:L_{2}
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 L_{2} 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
Copy Restrictions:
You
may make single copies of this document for your own personal use and for the
use of other students, but inclusion in another document, publication or any use
for profit requires my permission. Teachers may make multiple copies of this
document for their students if they first get permission from me. Merely send
me an email (Just click on Webmaster in the navigation bar.) with a onesentence
explanation of what you’re using the document for. I’ll give you permission in
a timely manner.
Making it Better: I
would be grateful if you would report any errors or suggestions for improvements
to me. Just click "Email Webmaster," site the item number, and tell me
your suggested change.
Printing Hint:
Most browsers will send both the navigation bar and the text to
the printer, and, as a result, will cut off the right edge of this document if
it the file is printed directly. To prevent this, highlight the
instructions portion only (not the navigation panel) and check "Selection" on
the Print dialog box; then click "Apply." This will eliminate the
navigation panel and get all of the instructions on the printed pages.
