 
Brief User Guide for TI83 Plus & TI84 Statistics
Contents: This document covers
single and twovariable statistics, scatter plot, regression analysis,
normalpdf, Student's Distribution, and much more.
INDEX:
To facilitate lookup, the instructions
are divided into the following categories:
I. Data
Manipulation  Entering data, sorting data, clearing lists, friendly values from
graphs.
II. SingleVariable Statistics  Histogram by hand, 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.
III. Two Variable Statistics – scatter plot, regression analysis,
finding r, r^{2}, a, and b in correlation using a
calculator, finding r, r^{2},
a, and b in correlation using a computation formula, testing the correlation
coefficient,
IV. Aids in doing statistics by hand
V. Permutations, combinations,
factorials, random numbers.
VI. Normal Distribution  Area under
a normal curve, Finding Z values, Graphing a curve, WINDOW
settings for graphing a curve, Probability Distribution Function using normalpdf(,
Graphing the
Normal
Distribution Using normalpdf(, normalcdf(, and ZInterval,
VII. Other Distributions  Finding a TInterval,
Student's t Distribution, Using invT to Find a tvalue given α
and df,
Chisquared Distribution,
binomialpdf, binomialcdf.
VIII. Hypothesis testing  mean and ztest
(data), mean and ztest (statistics), mean and ttest (data),
mean and ttest (statistics).
IX.
Statistics of two Populations  confidence interval for two dependent
population,
confidence interval for two
independent populations (Data and Stats),
X. Other Tests and
Inferences  oneway ANOVA, ChiSquare test for independence, X^{2 }Goodness of Fit,
APPENDIX: Simple program for calculating
inverseT with at TI83 Plus
RELEASE DATE: 10/1/06 DATE LAST REVISED:
9/17/10
© 2003
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.
NOW A WORD ABOUT MY
USE OF 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 is more
convenient to do arithmetic operations from the lists screenBut when
using a function such as
sum( , the operation
must be done from the home screen.
I.
Data Manipulation
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. 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 a
comma.
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 2nd,
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.
4) Friendly Values on Graphs Using TRACE:
Many times when you use the TRACE function,
you will get an xvalue such as 2.784532. If you
change the xrange in the WINDOW function to be a
multiple of 4.7, the xvalues will be "friendlier"
values that can be more easily plotted by hand.
Usually the easiest way to do this is to press ZOOM,
4, for ZDecimal and use Zoom In or Zoom Out
to adjust the window size if it's not satisfactory. That's
fine if you are satisfied with a symmetric window.
If you need an asymmetric window, you can get the friendly
values by pressing WINDOW
and setting the window parameters by hand. Let's take a value and say
that after a stat plot we get some "unfriendly" values and we press WINDOW and
get Xmin = .6 and
Xmax = 8.2. If we change Xmin to 0 and Xmax to 2*4.7 = 9.4; then we
will have friendlier values when
using TRACE.
II.
SingleVariable Statistics:
1) Graphing or Drawing a
Frequency Distribution Histogram (Ungrouped Data):
I have included
two methods for graphing a histogram or getting data to construct the histogram
by hand. 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.
Also I give a
procedure
for getting the
data from these graphs to draw a histogram by hand in the event that an
instructor requires that be
done.
Finally, I have included a calculator program 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 your
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:
a) Go to the lists and enter data. First, press
[STAT], [ENTER] to display the list tables.
b)
Enter the numbers in L_{1}. (Or whatever list you choose.)
c) Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
d) 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.
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.
C) Doing the Histogram
by Choosing Your Own Class Widths:
a) Go to the lists and enter data. First, press
[STAT], [ENTER] to display the list tables.
b)
Enter the numbers in L_{1}. (Or whatever list you choose.)
c) Find the
class width 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.
(2) Press STAT, ENTER and record the smallest number (the first one) and the
largest number (the last one).
(3) Let S represent the smallest data number,
L be the largest number, and C be the number
of classes you've chosen.
Find the class width, W, with the formula W =
(LS)/C. Round the number up to the next higher whole number.
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.
d) Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
e) 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.
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.
i) Press GRAPH and the new graph with the correct number of classes will
be displayed.
D) Using Either of These Methods to Get Data to Plot a 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 from
whichever of the graphs above that you choose.
a) 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.
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 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 your 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 as 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} .
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) 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) Press STAT, cursor to CALC and
press ENTER. "1Var Stats" will be displayed on the home
screen.
b) If your data is in list L_{1}
just press ENTER. Otherwise press 2nd and the list name where your
data is stored.
c) Cursor down and you will find
Q _{1} , Q_{3} , and Med listed. "Med" is the median.
12) Discrete Probability Distribution
Let's take a simple example to
demonstrate this: Suppose a word is flashed on a screen several
times while people are trying to
recognize the word. The list below indicates what percentage of the
group required a given number of flashes to
recognize the word.
No. of Flashes 1 2
3 4 5
Percent
27 31 18
9 15
P(x) .27 .31 .18
.09 .15
In summary, the method is to
enter the number of flashes into list L_{1} and the corresponding P(x)
values into L_{2} as
the frequency. The details are as follows:
a) Enter the number
of flashes in list L_{1} and the corresponding P(x) values in L_{2
}opposite the
number of flashes. (How to enter data in a list is covered at the
beginning of this document.)
b) Press STAT, cursor to
CALC and press ENTER. 1Var Stats will be displayed on the home
screen.
c) Press 2nd, L_{1},
press the comma, then 2nd, L_{2} . You should now have 1Var
Stats L_{1}, L_{2 }on the
home screen.
d) Press ENTER and the
values for the mean (expected value), standard deviation and other
statistics will be displayed.
e) If you need the
variance, merely reenter the value for the standard deviation,
σ_{x}^{
} , 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 σ_{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) 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
variance using this method would be this:
sum(L_{1}^{2}*L_{2})
 (sum(L_{1}
*L_{2}))^{2} .
14) 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.
15)
Finding the Standard Deviation and Variance of Ungrouped 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, 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. If the data is in L_{1},
just press ENTER, otherwise
press 2^{nd} and the list number where the data is stored
and then 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}^{ } ,
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.
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 TI82, TI83 Plus, or
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
expression “1Var Stats”
should be pasted to the home screen. If the data is in L_{1}, just
press ENTER, otherwise
press 2^{nd} and the list number where the data is stored.
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...
16.
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) 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. Press 2^{nd}, L_{1
}; then press the comma and finally
press 2^{nd}, L_{2}.
f) Press ENTER, and the standard deviation along with 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.
B. Calculating
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. Press 2^{nd}, L_{1 };
then press the comma and finally
press 2^{nd}, L_{2}.
c) Press ENTER and several statistics along with the standard deviation will be
displayed.
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.
17) 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.
18) 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. 1Var Stats
will be displayed on the home screen.
c) If the data are in lists L_{1} and L^{2} just press
ENTER. If they are in other lists, you must enter the lists. For
example, ^{ }
press 2^{ND}, L_{2}, comma, 2^{ND}, L_{3}
and then press ENTER
d) 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.
III. 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. LinReg (a+bx) will appear on the screen if you chose 8.
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 described above, just press
ENTER. If you have your data in other lists, you’ll need to enter the lists by pressing 2^{nd}, press the list
number for x_{,} comma, 2^{nd}, press the list number for y; then press ENTER. 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) If you want to graph the equation, then immediately after
LinReg(a+bx), enter the lists separated by
commas if the
lists are not in L_{1} and L_{2. }If the numbers are in
L_{1} and L_{2, }you need not
enter the list names.
f) Now, you want to store this as a Yvariable, say, Y1. So, do it this
way: Press [VARS], Cursor to
YVARS, [ENTER], [ENTER]. You should now have this on your screen.
LinReg Y1. (If the numbers
are in
other lists, the lists followed by commas are also included
g) Press [ENTER]. After a few seconds a long equation with
coefficients having several decimal
places will appear on the screen.
h) To graph that, 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) 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) 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 values 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
expression 2Var Stats, should be displayed on the screen.
c) If the data are in L_{1} and L_{2}, press ENTER and the
necessary values will be displayed. If the
data are not in those lists, you will have to enter the list numbers
where the data are stored.
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. On
the TI83 Plus or
TI84, a square root expression must be enclosed in parentheses.
Example: √(nΣx^{2} (Σx)^{2}).
Now let’s plug the numbers into the equation for r:
At this point you will save yourself a lot
of time if you calculate r and r² with the calculator. To do that,
press STAT, move the cursor to CALC and
press 8 for LinReg(a+bx. Press ENTER if the lists are in L_{1} and
L_{2}. If they are not in
those lists, you will need to enter the lists separated by a comma. Press
ENTER and
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.
6) 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 
5 
6 
7 
8 
Height x 
65 
65 
62 
67 
69 
65 
61 
67 
Weight y 
105 
125 
110 
120 
140 
135 
95 
13 
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. Press ENTER if the lists are in L_{1} and
L_{2}. If they are not in
those lists, you will need to enter the lists separated by a comma. Press
ENTER and
"a" and "b" 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.
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 the TI83
Plus or TI84, a
square root expression must be enclosed in parentheses. Example:
√(nΣx^{2} (Σx)^{2})
d) Enter the following formula on your worksheet, or in the calculator
if you're going to calculate the values.:
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 *65.125
=186.465…
7.
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.
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.
IV. Aids in doing statistics by hand.
General: Often in book problems in school you'll need to do a lot of
calculations by hand. These
techniques will save you a lot of arithmetic.
1. 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) Enter the list name you want to operate on by
pressing 2nd; then the list number, for example L_{1.
}c) 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}; then enter the corresponding frequencies
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.
"1Var Stats" will be pasted to the home screen. _{
}c) Press 2nd, L_{1}, 2nd, L_{2};
then 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 products
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}.
a) Enter 2nd, L_{1}, , 2nd, LIST.
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) Press (, 2nd, L_{1}, , 2nd, LIST.
b) 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 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.
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.
V. Permutations, combinations, factorials, random
numbers:
1. Finding 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.
2.
Finding 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.
3. Finding 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.
4. 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.
VI.
Normal Distribution:
Note:
In this section, a general method will be
outlined; then a specific example will be worked. The
same
problem will be used in several of the examples.
General, normalcdf(: This function returns the value of the area between two
values of the random variable
"x." This can be interpreted as the probability that a randomly selected variable will fall
within those two
values of "x," or as a percentage of the 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.
a) Press 2nd, DISTR, 2.
The term "normalcdf(" will appear on the home screen.
b) Enter the number for the
left boundary, right boundary, μ, and σ in that order. You do not need
to
close the parentheses, but it's okay if you do.
c) Press ENTER and the value of
the area between the two points will be displayed. Notice that
you do
not explicitly convert the points to 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(65,80,70,4.5.
c) Press ENTER and
you'll get 0.85361 which is, of course, 85.361 percent.
2.
normalcdf(: Area under a curve to the left of a point with μ (mean) and σ (std.
dev.) given.
Ex. 2: In the
above problem, determine the probability that the value is less than 62.
a) Complete
item a) in the general method above.
b)
Enter numbers so that your display is the following: normalcdf(1E9,
62,70,4.5. Notice that
the "" is a negative sign, not a minus sign. Enter "E" by pressing 2nd,
EE (The comma
key.)
c) Press
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: normalcdf(75,
1E9,70,4.5.
Enter "E" by pressing 2nd, EE (The comma key.)
c) Press
ENTER 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: normalcdf(72,
1E9,70,4.5/√(35)
Enter "E" by pressing 2nd, EE (The comma key.)
c) Press
ENTER 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 any Y= functions that may be active. Do this by moving the cursor
to a
highlighted = sign and pressing ENTER.
b) Press 2nd,
DISTR and cursor over to DRAW. Press 1 and ShadeNorm( will appear on the
home screen. Enter the correct parameters depending on whether the problem
is like 1, 2,
or 3 above.
c) 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.
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 ShadeNorm(
will appear on the
home screen.
c) Enter parameters so that your display looks like this: ShadeNorm(1E9,
62, 70, 4.5.
d) 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:
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 Ex. 1 immediately above, find the number x, such that a randomly selected number will be
below
that
number with a 90% probability.
a)
Press 2nd, DISTR, 3 to select invNorm(.
b)
Enter parameters so that your display looks like this: invNormal(.90,70,4.5.
c)
Press ENTER and your answer will be 75.766.
Ex.
2: Given a normal distribution with a mean of 100 and standard
deviation of 20. Find a value X_{o} such
that the given xvalue is below X_{o} is .6523. That is P(X<X_{o})
= .6523.
a) Press 2nd, DISTR, 3 to place "invNORM(" on the home screen.
b) Enter information so that the entry looks like the following:
invNORM(.6523,100, 20.
c) Press 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. to place "invNORM(" on the home screen.
b) Enter information so that the entry looks like the following:
invNORM(1.1,70, 12. 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 ShadeNorm(
will appear on the
home screen.
c) Enter parameters so that your display looks like this: ShadeNorm(1E9,
75.766, 70, 4.5.
d) 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 ShadeNorm(
will appear on the
home screen.
c) Enter parameters so that your display looks like this: ShadeNorm(40,
54, 46, 8.5.
d) Press ENTER and a reasonable looking graph should appear on the screen.
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 is used to find the fraction, and therefore
also the percentage, of the
distribution that corresponds to a particular value of x. The syntax of
this function is
normalpdf(X, μ, σ
A)
Finding the Percentage of a Single Value:
Ex. 1: Suppose that the mean of a certain distribution is 60 and the
standard deviation is 12.
What percentage of the population will have the value 50?
a) Press 2nd, DISTR, 1 to paste normalpdf( to the home screen.
b) Enter data so that your display is as follows:
normalpdf(50,60,12.
c) 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 gives the range within which the population mean is expected to fall
with a desired
confidence level. The sample size should be > 30 if the
population standard devation is not
known.
Ex. 1: Suppose we have a sample of 90 with sample mean x¯ =
15.58 and s = 4.61. What is the 95%
confidence level interval?
a) Press STAT, cursor to TESTS, and press 7.
b) On the screen that appears, cursor to "Stats" on the ZInterval screen and press ENTER.
c) Enter data opposite positions as follows:
σ: 4.61, x¯ :15.58, n:90, and CLevel: .95.
d) 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
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 7 to get the ZInterval screen.
c) Cursor to "Data" and press ENTER.
d) Next you need to know the sample standard deviation. To enter
that opposite σ, do this: Press 2nd, LIST,
move the cursor to MATH and press 7. The expression stdDev( will be pasted
opposite σ. _{
}
e) Press 2nd, L_{1 }, or whatever list you have your data in. When
you move the cursor the value will be entered.
f) Enter information as follows:
List: Press 2nd, L_{1}, Freq: 1, CLevel: .95.
g) Cursor to Calculate and press ENTER. The same type data will be
displayed as in Ex. 1 above.
VII. Other Distributions and Calculations:
1.
TInterval: If the sample size is <30, then the sample mean cannot be used for the
population mean, and
the ZInterval cannot be used. However, if the distribution is essentially normal, i.e.,
known to be normal
form other sources or has only one mode and is essentially symmetrical, then the Student t
Distribution
can be used.
Ex. 1: Suppose you take ten temperature measurements with sample mean x¯
= 98.44 and s = .3.
What is the 95% confidence level interval?
a) Press STAT, cursor to TESTS, and press 8.
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 to where 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 give that you have
20 data points.
a) Press 2nd, DISTR, 5, (6 on a TI84) to paste tcdf( to the home screen.
b) Enter data so that your display is as follows:
tcdf(1.4, 1E9,19.
c) Press ENTER and your answer should be .0888...
3. invT: Finding a tvalue Given
α and df:
If you are working a problem using the tvalue, there are different options
depending on your needs and whether
you're using a TI83 Plus or a TI84 Silver Edition.
TI84 Plus Silver Edition: This calculator has an invT, so do the
following:
(1) Press 2nd, DISTR, 4, and invT( will be pasted to the screen.
(2) Enter α or 1α, depending
on whether you have a left or right tail; then enter the degrees of freedom, df.
(3) Press ENTER and the value for "t" will be displayed.
Note that you may need to divide α by 2 if you
have not already made that adjustment.
TI83 Plus: This calculator does not have an invT, so you can do
either of two procedures:
(1) Look up the tvalue in your book. This is by far the easier.
(2) If you have an α that's not in the table or don't have a table,
you can do this:
Suppose you want the tvalue for α=.1 for a lefttailed test.
(a) Press MATH, 0, and the solver will be pasted to the screen.
(b) Press the UP arrow so that the equation is displayed.
(c) Press 2nd, DISTR, 5 and tcdf( will be pasted in as a formula.
(d) Enter data so that your entry will look like this: tcdf(1E9, X, 10) 
.100 and press ENTER.
(e) Press the UP arrow and enter 1 opposite X.
(f) Press ALPHA, SOLVE, and the value for "t" will be displayed
opposite X after about 20 seconds.
Suppose you want the tvalue for α=.1 for a righttailed test.
The steps are exactly the same except for these.
(d) Enter data so that your entry will look like this: tcdf(1E9, X, 10) 
.900 and press ENTER.
(e) Press the UP arrow and enter 1 opposite X.
Use a Calculator Program:
There are several program posted on the Web, for example, at
www.ticalc.org . I included a program
of
my own at the end of this document. It may not be the greatest, but it's
simple and it works.
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, 7 (8 for TI84), to paste χ^{2}cdf( to the home screen.
b) Enter data so that your display is as follows:
χ^{2}cdf(24, 1E9,19.
c) Press 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(12, .05, 4.
c) Press 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(12,
.05, 3.
d) Press ENTER and the answer, .0022364 will be displayed.
Longer Way:
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(12, .05, {4, 5,6,7,8,9,10,11,12}). Be sure
to use braces rather than parentheses.
c) Press STO, 2ND, L_{1} to tell the calculator which list to
store the individual values in.
Now, we want to also get the sum of all of these. Do that as follows:
d) Press ALPHA, : (the decimal point key); then 2ND, LIST, move the cursor to MATH, and press 5. The
expression
binompdf(12, .05, {4, 5,6,7,8,9,10,11,12}) : 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. If you need the individual numbers,
they 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(12, .05, 3.
c) Press 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. You will now have sum(binomialPdf(
posted to the home screen.
c) Enter data so that you have sum(binomialPdf(11, .9, {8,9,10})) on
the home screen. Be sure to use braces
rather than parentheses enclosing the numbers 8, 9, 10.
d) Press ENTER and .667...will be displayed.
VIII. Hypothesis Testing:
1. Testing for Mean and 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. stdDev(, will now be displayed
opposite σ. Now, enter you
list number where the dats is stored by pressing 2nd, and the list number, for
example L_{1} _{. }
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.
2.
Testing for Mean and 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.
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 for invNorm(.
b) Enter
α for a lefttailed or 1α for a righttailed and press ENTER.
c) The zvalue 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( will be pasted to the home screen.
b)
Enter numbers so that your entry looks like this: invNorm(.05, 5.25, .6.
For a left tail, enter the value
for α and for a right tail enter 1α..
c) Press
ENTER and the inverse 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.
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 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.
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.
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.
Simple Program for Calculating InverseT:
I have written
a simple program for those who
want to find tvalues with a calculator. Because the TI83Plus
has a fairly slow clock speed,
a solution may take 20 seconds or so. When
you enter the program, you can add more letters to the menu items if you
prefer. I have
abbreviated them to save memory space in my calculator.
The Program is in the Appendix at the end of this document.
IX.
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.
f) On the screen that appears, move
the cursor to "Data" and press ENTER; then enter 1 opposite Freq
and press ENTER.
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.
c) Enter the sample mean, standard deviation, and
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.
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.
X. 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.
Calculation for TI83 Plus:
What we are actually going to do is first find the sum of the values
listed opposite CONTRB on the TI84.
This sum will give us the value for X^{2}. We will then use the
X^{2} cdf to find the value for p.
a) Press STAT and enter the observed values in L_{1} and
the expected values in L_{2}.
b) Press 2^{ND}, QUIT to go to the home screen.
c) Press 2^{ND}, LIST and move the cursor to MATH. Press
5 to paste sum( to the home screen.
d) Enter information so that you have the following: sum((L_{1}L_{2})^{2}/L_{2}).
Press ENTER and you should get
the answer of 11.809 for X^{2}.
Calculate the pvalue:
f) Press 2^{ND}, DISTR, move th cursor to X^{2}cdf( and
press ENTER.
g) Enter information so that you have the following: X^{2}cdf(11.809,
1E9, 4). The “E” is made by pressing
2^{ND}; then the comma key.
h) Press ENTER and the value .018829…, the same value as with the TI84,
should be displayed.
XII: Appendix:
PROGRAM for calculating InverseT
: ”FKIZER 91906”
: INPUT “DF=”, D
: Menu(“SELECT”, Lft TL”, 1, “RT TL”, 2, “2TL”, 3)
:
Lbl 1
: solve(tcdf(1E9, X, D) – A, X, 1.7) →T
: Goto 4
: Lbl 2
: solve(tcdf(1E9, X, D) –(1 A), X, 1.7) →T
: GoTo 4
: Lbl 3
: solve(tcdf(1E9, X, D) – A/2, X, 1.7) →T
: Disp abs(T
:Lbl 4
:Disp T
Using the
Program:
a) After you’ve entered the
program, highlight the program name and press ENTER.
b) The program will ask for the confidence level, α, and then the
degrees of freedom, df. For this program,
α
is not divided by 2 when doing a twotailed test. Remember that for a
c) You will then be presented
with a menu to select either righttail, lefttail, or 2tail. Select the one
appropriate by
either pressing the appropriate number or highlighting the number and pressing ENTER. The answer will be
displayed in
approximately 20 seconds.
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
program. .
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your suggested change.
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