Brief User Guide for TI83 Plus Statistics
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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, 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.
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 formula.
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(, ZInterval,
VII. Other Distributions  TInterval,
invT Finding 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. Simple program for
calculating InverseT with a TI83Plus.
X.
Statistics of two Populations  confidence interval for two dependent
population,
confidence interval for two
independent populations (Data and Stats),
RELEASE DATE: 10/1/06 DATE LAST REVISED: 10/1/08
Back to home page.
NOTE: Copying limitations 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 for both use of a
calculator only and use of 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, and subtraction and storing the result in another
list, the operation can be done from
either the list screen or the home screen. Fro example L_{1}*L_{2}
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.) But when
using a function such as
sum( , the operation must be done from the home screen. So, I will be
using both the home screen and the list
screen to do list operations in this document.
I.
Data Manipulation
(NOTE: In some instances you may
want to clear a list or lists before you start entering data. You
can overwrite data already in a list, but remember that if the old list was
longer than the new one,
you must delete the remaining old data an item at a time. The easiest way
to clear one of the tabular
lists, 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 use more than one list separate
the lists by a comma.
1) Entering Data:
a) Press STAT; then ENTER. A table 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.
2) Putting Data
in Order:
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; then return to your
tables to view the sorted data. Note that you can also sort data
in descending order with
SortD.
3) 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 are satisfied with a symmetric window. If you need an asymmetric
window, you 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) Doing a
Frequency Distribution Histogram by Hand:
a) Use items 1 and 2 in Section I above to enter and sort your data.
b) Find the class width as follows:
(1) Let S represent the smallest data number (The first
number in your sorted list.), L be
the largest number (The largest number in the sorted
list.), and C be the number of
classes you've chosen. Find the class width, W, with the
formula W = (LS)/C.
c) Determine the limits of the classes by adding the class
width to each successive class.
Don't forget that the lower class limit is counted as part of
the class width.
d) Determine the number of data points in each class as follows:
(1) If your data is in L_{1}, go to that list. Make
sure your data is sorted in ascending order;
then scroll down to the last number that falls within the
upper limit of the first class. At
the bottom of the list your will see L1(#), where # is the
number of data items in your first
class.
(2) Scroll down to the last item of the second class and
subtract the number of items in the
first class from the number that appears in L1(#).
Continue this until you come to the
end of the list. Note that if you also want cumulative
frequency, just write down the
numbers as you progress.
e) Subtract 0.5 from each lower class limit of the first class to
get the lower boundary of the
first class. Add the class width to get successive boundaries.
f) Alternatively, you could do the histogram described below and
use the data classes and
values from that histogram.
2) Doing a Histogram with the TI83 Plus:
This procedure describes how to do a simple histogram for which
the calculator selects the class
width and, therefore, the number of classes..
First you need to get
your data into lists.
a) First go to the graphing screen by pressing the Y= button and
deselecting any functions so
that they won't be displayed with your graph.
Now, go to the list and enter data as follows:
b) Press [STAT], [ENTER]
c) Then enter the numbers in L1. (Or whatever list you choose.)
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 find the numbers for the limits of the classes and the number
of items in the class, press
[TRACE]; then use the cursor to move across the tops of the
bars in the histogram and read the
various numbers. .
3) Selecting Your Own Class Widths for the Histogram Generated by the
Calculator.
a)
Enter your data into List L_{1. } If your data is not in order,
you can sort it by pressing STAT,
selecting SortA(, then entering the list name of the data (often
L_{1}). As an example, you might
have this displayed on your screen: SortA(L_{1}. Now,
press ENTER, and your data will be sorted.
b) Now, from the sorted data, determine the class width and lower
boundary of the lowest class as
described under "Doing a Frequency Distribution by Hand" above.
Now do this:
c) Press [2nd], [STAT PLOT], highlight 1, and press [ENTER].
d) Highlight ON on the next screen; then highlight the histogram
symbol. Make sure L_{1} is entered
for the Xlist. Note that if you do not have tables data (
where frequencies are given), use the default
value, 1, for the frequency.
e) Press [WINDOW], and enter the lower boundary of the lowest class
as Xmin and your
chosen class width in Xscl. Note that once the lower
boundary and class width are set, the
upper limit is automatically determined. Set ymin at zero.
f) Press [GRAPH] and the histogram will appear. You can use
[TRACE] to display the value
of the boundary limits and frequencies of a particular bar on
the histogram.
g) If the graph extends above the top edge of the screen, Press
WINDOW and increase the Ymax
value. I also usually set Ymin to 1.
4. 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.
5. 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: Press 2nd, LIST, cursor to OPS, and press 6. cumSum(
will be posted to the home screen.
c) With the cursor after the parenthesis, press 2nd, L_{3, }
), STO, 2nd, L_{5, }ENTER. You will now have cumSum(L_{3})→L_{5}
pasted to the 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.
6) Relative Frequency polygon and Cumulative Relative Frequency
(Ogive) Graphs:
Do these exactly as in the frequency polygon and cumulative
frequency graph above except that after storing
the data (step b) for the frequency polygon) do this step: Press
2nd, L_{3 }/N, STO, 2nd, L_{3 }. This will convert
the data in L_{3 }to relative frequency.
7) 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 Ylist.
e) Press ZOOM, 9 and the histogram will be displayed.
Note: If you want to select your own classes do this 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.
2) Press GRAPH and the histogram will be displayed.
8) 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.
9) Cumulative Frequency (Ogive) Graph from Grouped Date:
a) Enter the upper class limits 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 }, do the following:
A) Press 2nd, LIST, cursor to OPS, and press 6. cumSum( will
be posted to the home screen.
B) With the cursor after the parenthesis, press 2nd, L_{2,
}), STO, 2nd, L_{4 }. You will now have
cumSum(L_{2}) →L_{4}
pasted to the 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.
10) Relative Frequency and Cumulative Relative Frequency Graphs for
Grouped Data:
Do these exactly as in the frequency polygon and cumulative
frequency graph above except that after storing
the data for the frequency polygon do this step: Press 2nd, L_{4
}/N, STO, 2nd, L_{4 }. This will convert the data
in L_{4 }to relative frequency. This assumes that the
frequency data is stored in L_{4 }.
N is the total number of data points.
11) 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) Press 2nd, QUIT to get out of the List.
c) Press 2nd, LIST, cursor to OPS, and press 6 to paste cumSum(
to the home screen.
d) Press 2nd, L_{2} , ), ÷ . You now should have cumSum(L_{2})/
on the home screen.
e) Press 2nd, LIST, cursor to MATH and press 5 to paste sum(
to the screen.
f) Press 2nd, L_{2}, ). You now should have cumSum(L_{2})/Sum(L_{2})
on the home screen.
g) Press x (the multiply symbol), 100, STO, 2nd, L_{3}. You
now should have
cumSum(L_{2})/Sum(L_{2}) *100→L_{3}
pasted to the home 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.
12) Box and Whisker Plot
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.
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.
13) 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.
14) Discrete Probability Distribution
Let's take a simple example to demonstrate this: Suppose a word is
flashed on a screen several
times while people are trying to recognize the word. The list below
indicates what percentage of the
group required a given number of flashes to recognize the word.
No. of Flashes 1 2 3 4 5
Percent 27 31 18 9 15
P(x) .27 .31 .18 .09 .15
In summary, the method is to enter the number of flashes into 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^{,
}
15) Doing a
Discrete Probability Distribution by Hand
Many teachers still see value in cranking out the numbers for these
statistics, so here are methods
to take some of the drudgery out of doing the arithmetic.
The 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) Press 2ND, L_{1}, x, 2ND, L_{2}, STO, 2nd, L_{3}.
You will now have L_{1}*L_{2}→L_{3} pasted to the home
screen.
b) Press ENTER and you will have the individual values stored in
list L_{3 }and displayed on the
home screen.
c) To get the sum of these values, do this.
(1) Press 2nd, LIST; cursor to MATH, and press 5.
The expression sum( will be pasted to
the home screen.
(2) Press 2ND, L_{1}
,x, 2ND, L_{2} , ), STO, 2ND, L_{3}.
You will have sum(L_{1} *L_{1})→L_{3}
pasted
to the home screen.
(3) Press ENTER and the sum of those values will be
displayed. Obviously if you only
need the mean and not the details of the
arithmetic, do only part c.
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_{3}, do the following:
a) Press 2ND, L_{1}, x^{2}, ,x, 2ND, L_{2}
, STO, 2ND, L_{3}. You will have L_{1}^{2}*L_{2}→L_{3
}pasted to the home
screen.
b) Press ENTER and the individual values will be entered in list
L_{3 }and pasted to the home
screen.
c) To get the sum of these values do the following:
(1) Press 2nd, LIST; cursor to MATH, and press 5.
The expression sum( will be pasted to
the home screen.
(2) Press 2ND, L_{1}
,x^{2} ,x, 2ND, L_{2} , ), STO, 2ND, L_{3}.
You will have sum(L_{1}^{2}*L_{2})→L_{3}
pasted
to the home screen.
(3) Press ENTER and the sum of those values will be
displayed and stored in L_{3}. Obviously
if you only need the sum of the values in the first
term and not the details of the arithmetic,
do only part c.
d) Now subtract
the value for µ^{2} from the last value obtained and that will be the
variance.
e) To obtain the standard deviation, take the square root of the
variance as follows:
(1) If you have just calculated the variance do press
2ND, √, 2nd, ANS, ENTER. Otherwise,
insert the value for the variance in place of
ANS.
NOTE: Obviously, if you only want to obtain the values for the
these three parameters, you can
use this method, but it is much easier to use method 14 above.
Just as information, the total
expression for the variance using this method would the this:
sum(L_{1}^{2}*L_{2})
 (sum(L_{1} *L_{2}))^{2}
.
16) 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 press 2nd,
QUIT to leave the lists.
b) Press 2nd, LIST and move the cursor to MATH.
c) Press 3 to paste mean( to the home screen.
d) Press 2nd, L_{1}, ), and then press the divide symbol.
e) Press 2nd, LIST, move the cursor to MATH, and press 7.
f) Press 2nd, L_{1} and then press ENTER to display the CV.
17. 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, then press
2^{nd} , QUIT to leave the tables.
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.
B. Calculating
Numbers to Plug into a Formula::
The standard deviation can be found easily by using 1Var
Stats as described above, but
many teachers require that students do the calculations by hand to learn
the details of the
process. The following give 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) (L_{1}) 
x^{2 (}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 
Σx˛=7492 
The variance
computation formula is as follows: s^{2} =
[(Σx˛ (Σx)˛)/n)]/(n1), where s^{2} is the variance . So, we will
need ∑x^{2} and ∑x to plug into the formula.
a)
Enter the data in the table as indicated previously in this document.
Press 2nd, QUIT to get
out of the lists.
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.
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 do the detailed
calculations, ship to filling in the formula in step “f.” Otherwise, continue
with the next step.
d) Now we’ll need an x^{2} column. Press 2^{nd}, L_{1},
x^{2}, STO, L_{2}. You should have
L_{1}^{2 }→L_{2} on the home screen.
e) Press ENTER and the numbers will be displayed on the home screen and
stored in list L_{2}.
f) Now, we want to use the number that we
previously recorded to plug into the variance
formula. So, at the home screen enter
(7492216^{2}/7)/(6).
g) Press ENTER and you should have 137.8…, which is the variance.
h) To find the standard deviation, press 2ND, √ , 2ND, Ans, ENTER, and
you will have
11.39...
18.
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 tems. 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 
Freq. (f) (L_{2)} 
3545 
40 
2 
4555 
50 
2 
5565 
60 
7 
6575 
70 
13 
7585 
80 
11 
68595 
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) Press 2ND, QUIT to get out of the list editor and press (, 2ND,
L_{1}, + 2ND, L_{2},), divide
symbol, 2 STO, L_{1}. You should have (L_{1}
+ L_{2})/2→
L_{1 }on the home screen. 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, then
press 2^{nd} , QUIT to leave the tables.
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}.
e) Press ENTER, and the standard deviation along with several other
statistics will be
displayed. The sample standard deviation is 14.868….
f) 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 Formula:
The standard deviation and variance for grouped 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 
Freq. (f) (L_{2)} 
xf 
_{ 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} =^{ } Σx˛f=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 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) Press 2ND, QUIT to get out of the list editor and press (, 2ND, L_{1},
+ 2ND, L_{2},), divide
symbol, 2 STO, L_{1}. You should have (L_{1} + L_{2})/2→ L_{1 }on the home screen. Press
ENTER
and the midpoints will be stored in L_{1}.
b) Press STAT, ENTER to go to the lists and store the frequencies in list L_{2}.
After you have
finished entering the frequencies and midpoints, press 2^{nd}, QUIT
to leave the lists.
Now let’s calculate the required numbers.
c) 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}.
d) 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=3780,
∑x^{2}=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.
e) Calculate xf and store it in
L_{3} by pressing 2ND, L_{1}, *, 2ND, L_{2}, STO, L_{3.}
You should have
L_{1*}L_{2}→L_{3 }on the home screen. Press
ENTER and the products will be stored in list L_{3 } and will
be displayed on the home screen.
f) Calculate x^{2}f by pressing 2ND, L_{1, }x^{2} , * ,
2ND, L_{2}, STO, 2ND, L_{4} . You should now have
L_{1}^{2}_{ }*L_{2}→L_{4} on the
home screen.
g) Press ENTER and the results will be stored in list L_{4 } and will
be displayed on the home
screen.
h) You
don’t need to calculate
Σf. That is the value for “n” that you previously recorded.
i)
You don’t need to
calculate
Σxf. That is the value for ∑ x that you previously recorded.
j) Now,
you want to plug the appropriate numbers into the formula for the variance. From
the
home screen enter
(2966003780˛/50)/(49)
k) Press ENTER and you should have 221.06, which is the variance.
l) If you want the standard deviation, press 2ND, √ , 2ND, Ans, ENTER, and you
will have 14.868...
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. (If your data is not in order you can sort in
order by pressing [STAT], select 2,
SortA( or SortD( for descending order. SortA( , or SortD( will be
posted to the home screen. Press
[2nd], L_{1}, 2nd, L2, [ENTER]. BE CAREFUL! If your data in
L_{2 }is not in ascending order when
correlated to L_{1, }then your data in L_{1} and L_{2}
will not be correlated correctly after sorting.
d) Press [2nd], [STAT PLOT] and press [ENTER] to turn Plot 1 on.
e) 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.
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 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), press [2nd], [L1] [,],
[2nd], [L2], [, ]. ( Note that the commas outside the brackets are
just separators so you can tell what
I'm doing. They do not appear in the syntax.
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.
QuartReg L1, L2, Y1
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 to get a good display.
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 press GRAPH to graph it.
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 Formula:
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:
d) r= (nΣxy
–ΣxΣy)/[(√nΣx^{2} (Σx)^{2})(√nΣy^{2} (Σy)^{2})]
= (8*62750521*960)/(√(8*33979521^{2})(√(8*116900960^{2}))
=.7979…..
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.
6) Finding the Values a and b for the
BestFit Equation^{ }Using a Formula:
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 
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 these
parameters: ). So, you will need to record these values:
xbar=65.125, Σx=521, Σx^{2}=33979,
n=8, Σy=960, ybar=120, Σy^{2}=116900, Σxy=62750
c) Plug these numbers into the formula and then enter the expression 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 values in the calculator for this formula:
b=(nΣxy
–ΣxΣy)/(nΣx^{2} (Σx)^{2}).
=(8*62750521*960)/(8*33979521^{2})
=4.7058…..
e) Now, calculate the value for a from the formula:
a= ybar –b(xbar)
=1204.7058 *65.125
=186.465…
f) 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, A.
2) Enter the
denominator and store it in variable D. 8*33979521^{2} , STO,
ALPHA, D.
3) Enter N÷D
and press ENTER.
You'll get the same answer as above.
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.
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.
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,
before pressing ENTER in item a, press 2nd,
L_{1}, STO, 2nd, L_{3. }Then press ENTER.
c) Obviously, xy can be obtained by merely substituting the subtraction
symbol for the
multiplication symbol in atep a) above.
4. Squaring operations such as elements
of lists.
a) To square the elements of a data set, first enter the data in a list,
for example 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},
then before pressing ENTER in
item b above, press 2nd, STO, 2nd, L_{3}. 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 xxŻ (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},
then before ENTER in item "c" above, press
STO, 2nd, L_{3}; then ENTER
6. Finding (xxŻ )^{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},
before pressing ENTER in item "c"
above, press STO, 2nd, L_{3}; then 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.
k) Press ENTER and you should have 138.1428, which is the variance.
l) If you want the standard deviation, press 2ND, √ , 2ND, Ans, ENTER, and
you will have 11.75...
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}.
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 1 and
a standard deviation of 0. I use the
values 1E9 and 1E9 for left or right tails. The E in obtained by
pressing 2nd, EE. This can be used to solve
such problems as the following: P(x<90), P(x>100), or
P(90<x<120). If 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. 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.
5. invNorm(:
Inverse Probability Calculation:
Find the number x, in a
normal distribution such that a number is less than x with a given
probability. The 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.
6. 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.
7. 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, 8.2.
d) Press ENTER and a reasonable looking graph should appear on
the screen. The area
under the curve, 0.603198, will be displayed on the screen
along with the upper and lower
bounds.
8. 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.
9. 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 xŻ
= 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, xŻ :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., know 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 xŻ = 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: xŻ :98.44, S_{
x} : .3_{ }n:10, and CLevel: .95.
d) Cursor down to "Calculate", press ENTER, and, after a few
seconds, the interval (98.228, 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, 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 will also be
posting a
program that I have written some time soon. It may not be
the greatest, but 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, to paste χ^{2}cdf( to the
home screen.
b) Enter data so that your display is as follows: ^{χ2}cdf(24,
1E9,19.
c) Press ENTER and your answer should be .1961...
7. 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.
8. 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( 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(
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 CALC 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.
IX. Simple Program for Calculating InverseT:
This is 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.
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:
: ”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
X.
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, press 2nd, QUIT to go to the home screen.
Now, store the paired differences in list L3 as follows:
c) From the home screen, press 2nd, L1, minus sign, 2nd, L2.
d) Press STO, 2nd, L3. You should now have L1  L2 → L3 on the home
screen.
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.
b) After entering the data, press 2nd, QUIT to go to the home screen.
To go to the confidence interval screen do this:
c) Press STAT, move the cursor to TESTS, and press 0 (zero).
d) On the screen that appears, move the cursor to Data and press ENTER.
f) Opposite "List 1," press 2nd, L_{1} and opposite "List2,"
press 2nd, L_{2}.
g) Set the confidence level you choose opposite "CLevel."
h) Highlight "No" opposite "Pooled" if there are no assumptions about
the variations.
i) Move the cursor to "Calculate" and press ENTER. The confidence
interval along with other statistics will be
displayed.
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