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Brief User Guide for TI-82 Statistics
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. Single-Variable 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,
standard deviation with a computation formula.
III. Two Variable Statistics – scatter plot, regression analysis,
finding r, r2, a, and b in correlation using a
calculator, finding r, r2, a, and b in correlation using a
computation formula.
IV. Aids in doing statistics by hand
V. Permutations, combinations,
factorials, random numbers.
VI. Binomial Distribution, pdf, cdf.
VII. Probability with the Normal
Distribution:
Notice that the TI-82 does not have the
capability to do distributions and inferential statistics as does the
TI-83 Plus and TI-83.
RELEASE DATE: Beta version - 11/5/08 DATE LAST REVISED:
11/5/08
Printer friendly page here.
IMPORTANT NOTICE: This is a beta version, meaning that I have not checked
it all out on my calculator. Please
use it with care.
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. For example L1*L2
will do the same thing at the list screen as
L1*L2→L3 at
the home screen. (The arrow is a result of pressing STO.) 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. DO NOT
press the DEL key or you will delete the list entry. You can also clear a number of lists or any list as
follows:
a) Press STAT, 4 (ClrList). This will paste "ClrList" to the home screen. Press 2nd;
then the button
for the list number you want to clear, for example
L1
; then press ENTER. If you want to clear more
than one list, separate the lists by a
comma.
1)
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.
2) Putting Data in Order:
a) Press
STAT, 2 (SortA). This will paste SortA to the home screen.
b) Press 2nd,
L1 (or whatever list you want to sort); then press ENTER. Finally, return
to your
tables to view the sorted data. Note that you can also sort data in descending order with
SortD.
3) Friendly Values on Graphs Using TRACE:
Many times when you use the TRACE function,
you will get an x-value such as 2.784532. If you
change the x-range in the WINDOW function to be a
multiple of 4.7, the x-values 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 X-min = .6 and
X-max = 8.2. If we change X-min to 0 and X-max to 2*4.7 = 9.4; then we
will have friendlier values when
using TRACE.
II.
Single-Variable 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 =
(L-S)/C. Round
the number up to the nest higher whole number.
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 L1, 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 with the calculator as described below and use the data
classes and
values from that histogram.
2) Doing a Histogram
with the TI-82:
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) Go to the graphing screen by pressing the Y= button and
deselecting or clearing 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]
, [ENTER] to turn Plot 1 on.
e) Cursor to
the icons opposite Type, select the fourth icon, histogram, and press [ENTER] to
highlight the histogram icon.
f) Enter L1
(or whatever list your data is in) opposite Xlist, by pressing 2nd, L1.
Make sure that
1 opposite Freq is highlighted if you have ungrouped data.
g) Press [ZOOM]; then 9 (ZoomStat)
and the histogram will
appear on the screen. You may need
to press WINDOW and adjust the window size to get the best display.
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 L1. 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 L1).
As an example, you might
have this displayed on your screen: SortA(L1.
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,
the fourth symbol..
Make sure
L1
is highlighted for the Xlist.
Note that if you do not have tables data ( where frequencies are given),
highlight 1 opposite Freq.
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 x-min, x-max, and "n"
on a sheet of paper in tabular form.
b) Add one-half the class width to each x-min value and record those
values. Store these values in a list, for example
L2 if you have your histogram data in L1. Store the
corresponding values of "n" in L3.
c) Press 2nd, STAT PLOT, ENTER. If "On" is not highlighted; then
select it and press ENTER.
d) Highlight the second icon;
then highlight L2
opposite Xlist and L3 opposite Ylist.
e) Press ZOOM, 9 and the graph will appear on the screen.
NOTE: Some teachers or texts prefer return-to-zero graphs. If your
course requires that, do the following after step b)
above:
A. Calculate a
midpoint of a new class preceding the first class
and another midpoint after the last class. These
values will be entered into L2. To do that place the
cursor at the first item in L2, press INS and replace the zero that
appears with your the first midpoint you calculated. Go to the
bottom of the L2 list and enter the second value you
calculated.
B. Now you want to enter zero in L3 opposite each of these
new midpoints. Place the cursor at the top of L3 and press
INS. A zero will be added. Now cursor to the bottom of the list
and enter a zero opposite the last new midpoint
that you entered in L2.
C. Proceed with step c) above.
5. Constructing a Cumulative Frequency Chart (Ogive) Graph:
a) Enter the Xmax values that you recorded above in a list. For
example, L4 if you still have data in the
other lists.
b) Next we want to
store the cumulative frequencies in L 5, but the TI-82 has no cumSum
function. So, we have three
options: Either do them by hand, use an expression, or use a program. I'll first show a
method that I have given
students to do the cumulative frequencies by hand; then at the end of this procedure, I'll show
give you a short
calculator program
and also how to do it using an expression. Either of the latter two
methods will store
the cumulative frequencies in a list. First doing them by
hand we'll assume that your frequencies
are stored in L4 , and your want to store the cumulative frequencies
in L5.
c) From the home screen
press 2nd, L4 , STO, 2nd, L5 to store the list 4 data in L
5. Then press 2nd, LIST, ENTER
to go to the lists.
d) Let's assume for simplicity that
we have 1, 2, 5 , 6 stored in L5. Place the cursor on 2 and
enter 2+1 to get 3 in
the
second item. Move down to 5 and enter 5+3 to get 8 as the third item.
Next move down to 6 and enter 6+8
to get
14. Continue with this method until you have completed the list.
e) Now that you have the
cumulative frequencies in L5, press 2nd, STAT PLOT; then press
ENTER, ENTER to
turn on the plot function.
f) Highlight the second icon;
then highlight L4
opposite Xlist and press ENTER. Highlight L5 opposite Ylist
and press ENTER. .
NOTE: If you did a return-to-zero graph for the
frequency polygon, go to the list and delete the last
midpoint and zero in L4
and L5 respectively.
g) Press ZOOM, 9 and the graph will appear on the screen.
Now the expression:
(1) First we want to store the first number in L4 in the first
position of L5. Do that by pressing 2nd, L4(1), STO,
2nd,
L5, (, 1, ). You should now have L4(1)→
L5(1). Press ENTER to store the number.
Alternatively, you cold just
go to the list and enter the number.
(2) Now, we want to store 0 in the variable X. From the home
screen, press 0, STO, X,T,O, ENTER.
(3) Finally, enter the following expression: X+1, STO, X, ALPHA, : L5(X)
+L4(X+1), STO, L2(X+1). You should have
the following expression on the home screen: X+1→X:L5(X)
+L4(X+1), →, L2(X+1).
Press ENTER continuously
until the an error message appears. That error signals that you have
reached the end of the list. Note that you
can change the lists if your list data is not in the list I have used. I
used L4 and L5 only because I was continuing
the problem above.
(4) Now go back to item "e" to do the plotting.
Now for the
short program:
Note that I have written this for lists 1 and 2. You
can change it to anything you
wish.
CUMSUM (This is the name I gave the program)
dim L1 →N
N→dim L2
L1(1)→.L2(1)
For (K, 1, N-1)
L1(K+1) +L2((K)→L2(K+1)
END
Now go back to item "e" to do the plotting.
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 they require
relative frequency data obtained as described in the Cumulative Frequency section.
7) Histogram Using Grouped Data:
a) Enter the midpoints of the classes into L1 and the
corresponding frequencies into L2 .
b) Press 2nd, STAT PLOT, ENTER.
c) If "On" is not highlighted, select it and press ENTER.
d) Move the cursor to the histogram symbol and press ENTER; then
enter L1 opposite Xlist and L2 opposite 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, L3
if you have data in the first two lists.
b) If you have the frequency in L2 , do the following:
Use either of the three methods described at the end of item 5 above for
the cumulative frequency.
c) Press 2nd, STAT PLOT, ENTER. If "On" is not highlighted, select
it and press ENTER.
d) Highlight the second icon, and enter L3 opposite Xlist
and L4 opposite Ylist.
e) Press ZOOM, 9 and the graph will be displayed.
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, L4 /N, STO, 2nd, L4 . This will convert the data
in L4 to relative frequency. This
assumes that the frequency data is stored in L4 .
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 L1 and the corresponding frequencies in L2.
If you want the graph to start
at zero, enter the first lower boundary with zero for the frequency.
b) Press 2nd,
QUIT to get out of the List.
c) You
must now get the cumulative sum of each term and divide it by the sum of L2,
multiply that by 100 and
store that result in L3. It's probably easier to just calculate
the sum of L2 than to try to enter that into the string.
To simplify matters, let's just store that sum in the variable A. To do
that, press 2nd, LIST, 5(sum), 2nd, L2, STO,
ALPHA, A, ENTER.
and record the number for future use.
Now the expression:
(1) First we want to store the first number in L2/A in the
first position of L3. Do that by pressing 2nd, L2(1),
÷100,
STO, 2nd, L3. You should now have L2(1)/100→L3(1).
Press ENTER and the number will be stored.
(2) Now, we want to store 0 in the variable X. From the home
screen, press 0, STO,
the key [XTOn], ENTER.
(3) Finally, enter the following expression: X+1, STO, X, ALPHA, : L5(X)
+L4(X+1), STO, L2(X+1). You should have
the following expression on the home screen: X+1→X:(L2(X)
+L3(X+1))/A*100, →, L3(X+1).
Press ENTER
continuously until the an error message appears. That error signals that you have
reached the end of the list.
Note that you can change the lists if your list data is not in the list I have used. I
used L4 and L5 only because
I was continuing the problem above.
Plotting the Data:
d) Press 2nd,
STAT PLOT, ENTER
e) Select the
second icon and enter L1 opposite Xlist and L3 opposite
Ylist.
f) Press
ZOOM, 9 and your graph will be displayed.
g) You can
find the exact percentiles of the boundaries by using TRACE, and approximate
percentiles of
other x-values 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
box-and-whisker plot,
icon 3, and
press [ENTER] to highlight the box plot icon.
f) Enter the list you put the data in, usually L1, in the Xlist, by pressing 2nd, L1.
or whatever list
you chose. Highlight
1 for the Freq.
g) Press [ZOOM]; then 9 (ZoomStat)
and the box-and-whisker plot will
appear on the screen.
h) To find the numbers for the limits of the quartiles, press [TRACE]; then use the cursor to move
across
the diagram and obtain the values for quartiles or the beginning and ending
values.
13) Box and Whisker Plot by Hand
You can save yourself considerable
calculation if you use the calculator to find Q1, Median, and Q3
when doing a box-and-whisker plot by hand.
To find those values do the following:
a) Press STAT, cursor to CALC and
press ENTER. "1-Var Stats" will be displayed on the home
screen.
b) If your data is in list L1
just press ENTER. Otherwise press 2nd and the list name where your
data is stored.
c) Cursor down and you will find
Q 1 , Q3 , 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 L1 and the corresponding P(x)
values into L2 as
P(x). The details are as follows:
a) Enter the number
of flashes in list L1 and the corresponding P(x) values in L2
opposite the
number of flashes. (How to enter data in a list is covered at the
beginning of this document.)
b) Now let's find the
mean: Remember that the mean is this: mean =
Σxp(x).
c)
Press 2nd, LIST, move the cursor to MATH and press 5(sum). Press (,
2nd, L1,*, 2nd, L2, ). You
should have this expression: sum( L1* L2)
d) Press ENTER and
the mean, 2.54, will be displayed.
e) Now let's do the
variance. Remember that the formula for the variance is this:
Σx2 P(x) -
µ2
or in terms of xP(x) this:
Σx2 P(x) -
(Σxp(x))2 .
f)
Press 2nd, LIST, move the cursor to MATH
and press 5(sum). Press (,
2nd, L1, x²,x, 2nd, L2, ) -
(, 2nd, LIST, move the cursor to MATH and press 5.
You should now have this on the screen
sum(L12*L2)
- (sum(
g) Press
2nd, L1, x, 2nd, L2, ),)² . You should now have the
following on the screen:
sum(L12*L2)
- (sum(L1
*L2))2
h) Press ENTER and
the variance, 1.868..., will be displayed.
i) If you need the
standard deviation do this: Press 2nd, √,
2nd, Ans, ENTER. The value, 1.3668... will
be displayed.
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 L3, do the following:
(The x-values should
should
be stored in L1 and the p(x) values in L2.)
a)
Highlight the title for L3 and press 2ND, L1, x, 2ND, L2.
You will now have L1*L2 in the left bottom of
the
list screen.
b)
Press ENTER and you will have the individual values stored in list L3.
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, L1 ,x, 2ND, L2 , ), STO, 2ND,
L3. You will have sum(L1
*L1)→L3 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:
Σx2 P(x) -
µ2 where µ is the mean obtained as above. To obtain the
individual
values
of the first term,
x2 P(x).
and store them in list L3, do the following:
a) From the list screen, place the cursor on the title for list L3 ,
press 2ND, L1, x2, ,x, 2ND, L2. You will
have L12*L2
at the bottom left of the lists.
b) Press ENTER and the individual values will be
entered in list L3.
c) To get the sum of these values do the following:
(1) From the home screen, press 2nd, LIST; cursor to MATH, and press 5. The expression sum
will be
pasted to
the home screen.
(2) Press (, 2ND, L1 ,x2 ,x, 2ND, L2
, ), STO, 2ND, L3. You will have sum(L12*L2)→L3
pasted
to the home screen.
(3) Press ENTER and the sum of those values will be displayed and stored
in L3. 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 µ2 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(L12*L2)
- (sum(L1
*L2))2 .
16) Calculation of Coefficient of
Variation from List Data:
The coefficient of
variation, CV=s/x-bar, is a simple arithmetic calculation if you have the mean
and standard deviation.
But calculations from a list are a little more involved. Here's an easy
way
to do it.
a) Store the data in a
list, for example L1, and press 2nd, QUIT to leave the lists.
b) Press STAT, move
the cursor to CALC. Press ENTER and "1-Var Stats" will be displayed on the home
screen.
c) Press ENTER and the
statistics will be displayed on the screen.
d) Calculate the variance
as follows:
(1) Enter the given value
for the mean from the list of statistics, press divide, enter the value for
Sx
or σx whichever is appropriate.
e) Press ENTER and the CV will
be displayed.
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) Press STAT, move the cursor to CALC, and press ENTER. The expression
“1-Var Stats”
should be pasted to the home screen. If the data is in L1,
just press ENTER, otherwise
press 2nd
and the list number where the data is stored and press ENTER. In either
case,
the standard
deviation and several other statistics will be displayed.
B. Calculating
Numbers to Plug into a Computation Formula::
The standard deviation can be found easily by using 1-Var
Stats as described above, but
many teachers require that students do the calculations by hand to learn
the details of the
process. The following give a method for using the TI-82, TI-83 Plus, or
TI-84 for doing much
of the arithmetic required and obtaining numbers to plug into the formulas.
Suppose that students did sit-ups according the table shown below.
|
Student |
Sit-ups (x) (L1) |
x2 (L2) |
|
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: s2 =
[(Σx² -(Σx)²)/n)]/(n-1), where s2 is the variance .
So, we will
need ∑x2 and ∑x to plug into the formula.
a)
Enter the data in the table as indicated previously in this document.
b) Press STAT, move the cursor to CALC, and press ENTER. The
expression “1-Var Stats”
should be pasted to the home screen. If the data is in L1, just
press ENTER, otherwise
press 2nd and the list number where the data is stored.
c) Copy n=7, ∑x = 216, and ∑x2 =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 x2 column. Highlight the title
on list L2, and press 2nd, L1,
x2. You should have
L12 at
the bottom left of the list screen.
e) Press ENTER and the numbers will be in list L2.
f) Now, we want to use the number that we
previously recorded to plug into the variance
formula. So, at the home screen enter
(7492-2162/7)/(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
Midpoint x (L1) |
Freq. (f) (L2) |
|
35-45 |
40 |
2 |
|
45-55 |
50 |
2 |
|
55-65 |
60 |
7 |
|
65-75 |
70 |
13 |
|
75-85 |
80 |
11 |
|
685-95 |
90 |
11 |
|
95-105 |
100 |
4 |
c)
Enter the class midpoints in list L1. You
can either do the midpoints by hand or calculate
and store them in list L1 as follows:
(1) Store the lower boundaries in list L1 and the upper
boundaries in L2.
(2) Place the
cursor on the title of L1; then press (, 2ND,
L1, + 2ND, L2,), ÷, 2 . You should
have (L1
+ L2)/2
at the bottom left of the tables. Press ENTER and the midpoints will be
stored in L1.
d) Enter the frequencies in L2 as
described under Entering Data immediately above, then
press 2nd , 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
“1-Var Stats”
should be pasted to the home screen. Press 2nd, L1
; then press the comma and finally
press 2nd, L2.
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 value, pressing
the x2 button, and then ENTER.
B. Calculating
from Grouped Data to Plug into a Computation Formula:
The standard deviation and variance for grouped are similar
to ungrouped data except that the
x-values are replaced by the midpoints of the classes. Let's assume some
sort of grouped
data as indicated by the first and third columns below.
|
Classes |
Class
Midpoint x (L1) |
Freq. (f) (L2) |
xf
(L3) |
x2f
(L4) |
|
35-45 |
40 |
2 |
80 |
3200 |
|
45-55 |
50 |
2 |
100 |
5000 |
|
55-65 |
60 |
7 |
420 |
25200 |
|
65-75 |
70 |
13 |
910 |
63700 |
|
75-85 |
80 |
11 |
880 |
70400 |
|
685-95 |
90 |
11 |
990 |
89100 |
|
95-105 |
100 |
4 |
400 |
40000 |
|
|
|
n=Σf=50 |
∑x=Σxf=3780 |
∑x2 = Σx²f=296600 |
The formula for the
grouped data variance is this:
s2 =(
Σx2 -(Σxf)2 /Σf)/(Σf-a)
a) You can either do the midpoints by hand or calculate and store them in list L1
as follows:
(1) Store the
lower boundaries in list L1 and the upper
boundaries in L2.
(2) Place the cursor on the title of L1;
then press (, 2ND,
L1, + 2ND, L2,), ÷,
2 . You should have (L1
+ L2)/2
at the bottom left of the tables. Press
ENTER and the midpoints will be stored in L1.
b) Press STAT, ENTER to go to the lists and store the frequencies in list L2.
After you have
finished entering the frequencies and midpoints, press 2nd, 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 “1-Var Stats”
should be pasted to the home screen. Press 2nd, L1 ;
then press the comma and finally
press 2nd, L2.
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=∑xf=3780,
∑x2=∑x2f=296600, and n=50. You’ll need these values later.
Notice that the value for ∑f is listed as n in the calculator and ∑xf is
listed as ∑x and ∑x2f is
listed as ∑x2.
NOTE: You now have enough numbers to plug into the formula and solve for the
variance.
If you are not required to do the detailed calculations to fill in the table,
skip to item “j” below.
Otherwise continue with the next step.
e) Calculate xf and store it in
L3. Highlight the title for L3 and press 2ND, L1, *, 2ND, L2..
You
should have L1*L2 at the bottom left of
the list screen. Press
ENTER and the products will be stored
in list L3.
f) Calculate x2f and store it it L4. Highlight
the title for L4 and press 2ND, L1, x2 , * ,
2ND, L2. You should
now have L12 *L2
at the bottom left of the list screen.
g) Press ENTER and the results will be stored in list L4 .
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
(296600-3780²/50)/(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. Two-variable Statistics
1)
Scatter Plot
First you need to get your data into lists.
a) Go to the graphing screen by pressing the Y= button and
deselecting any functions so that
they won't be entered on your graph.
If you want to clear the lists before entering data, see the
note at
the beginning of this document.
b) Press
[STAT], [ENTER] to go to the list tables.
c)
Enter the data-point numbers ( the x-values) in L1 and the
corresponding values (y-
values)
in L2.
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) Highlight L1 opposite Xlist, and L2
opposite 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 x-y line chart
Do that the same
as the scatter plot in item 1 above except that when you select the type, choose
the
second icon for the line symbol rather that the scatter-diagram icon.
3) Regression Analysis:
Assume that you have the
following information on the heights and weights on a group of young
women:
|
|
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, x-values, in L1 and
the corresponding values in L2.
b) After you have finished entering data, Press[STAT].
c) Cursor to CALC and press <9>, [ENTER] (Where <9> is just the number
9
from the keyboard.)
LinReg (a+bx) will appear on the screen if you chose
9. Note that if you want to use QuadReg
or some other analysis, press
the number to the left of that entry.
d) If you want to graph the equation of the best-fit line, ship to item
“e” below. If you have your data
in the L1 and L2 as described above, just press
ENTER. If you have your data in other lists, you’ll
need to enter the lists by pressing 2nd, press the list
number for x, comma, 2nd, press the list number
for y; then press ENTER. In either case a, b, and r
will be displayed.
ANSWER: If you pressed ENTER you should have these values:
a=-186.47.., b=4.705…,
and r=.7979…, if
you want r2 , just enter the value for r and press x2 ,
then ENTER. You will get
r2
=.63366….
e) If you want to graph the equation, press
Y=, VARS, 5 (for Statistics), move the cursor to EQ, and press 7
(for RegEQ. The
regression equation will be entered opposite Y1=.
f) Press ZOOM, 9 (for ZoomStat) and
the graph will be displayed.
g) If you also want to graph the
scatter plot on the same screen as the line graph, do the following:
1)
Press 2nd, STAT PLOT; then press ENTER, ENTER.
2) On
the screen that appears, select the first icon and press ENTER.
3) Move
the cursor to L1 opposite Xlist and press ENTER. Next,
move the cursor to L2 opposite
Ylist and press ENTER.
4) Select
your favorite mark is you have one; then press ZOOM, 9 and the scatter diagram
and the
best-fit line will be displayed.
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) Move the cursor opposite Xlist,
highlight L1 and press ENTER. Move the cursor opposite Ylist,
highlight L2
and press ENTER. Select the mark you choose if you wish. (I like a + ).
d) Press ZOOM, 9 (for ZoomStat) and the scatter plot and best-fit graph
will appear on the screen.
e) You can press [TRACE] to display the x-y values of the data points, or
press the down arrow to
jump to points on the line.
Note that if your data has several decimal places and you'd rather have fewer,
you can make the data
friendlier by making the x-distance (xmax-xmin) a multiple or sub-multiple of
9.4.
5) Finding the Correlation Values r and r2
Using a Computation 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, x-values, in L1 and
the corresponding values in L2.
NOTE: The formula for “r” is this: (nΣxy –ΣxΣy)/[(√nΣx2- (Σx)2)(√nΣy2-
(Σy)2)]. So, you will
need Σx, Σy, ΣxΣy, Σx2, Σy2,, and n. You can
get all of these by using the 2-Var Stats
function. Use that as follows:
b) With the data in lists L1 and L2 press STAT,
move the cursor to CALC, and press 2. The
expression 2-Var Stats, should be displayed on the screen.
c) If the data are in L1 and L2, 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, Σx2=33979, n=8,
Σy=960, Σy2=116900, Σxy=62750.
NOTE: Just a few words on entering the data in the calculator: All
denominators and
numerators with more than one term must be enclosed in parentheses. On
the TI-82, a square root
with more than one term must be enclosed in parentheses.
Example: √(nΣx2- (Σx)2).
Now let’s plug the numbers into the equation for r:
d) r= (nΣxy
–ΣxΣy)/[(√(nΣx2- (Σx)2)(√(nΣy2- (Σy)2)]
= (8*62750-521*960)/(√(8*33979-5212)(√(8*116900-9602))
=.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*62750-521*960, STO, ALPHA, N.
2) Calculate the
denominator and store it in two separate variables M and D. In this manner
√(8*33979-5212 ) , STO, ALPHA, M; then √(8*116900-9602),
STO, ALPHA, D.
3) N÷(M*D), ENTER.
You'll get the same answer as above.
6) Finding the Values a and b for the
Best-Fit Equation Using a Computation 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)/(nΣx2- (Σx)2). So, you will need to record the
values
for . x-bar, y-bar, Σx, Σy, ΣxΣy, Σx2, Σy2, and n..
You can get all of these by using the 2-Var Stats function.
Use that as follows:
a) With the data in lists L1 and L2 press STAT,
move the cursor to CALC, and press 2. The
expression 2-Var Stats, should be displayed on the screen.
b) Press ENTER and the necessary values will be displayed. Notice that
you will need to
scroll down to get some of the values on the screen.
(Record the
values for these
parameters: ). So, you will need to record these values:
x-bar=65.125, Σx=521, Σx2=33979,
n=8, Σy=960, y-bar=120, Σy2=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 TI-82, a
square root expression must be enclosed in parentheses. Example:
√(nΣx2- (Σx)2)
d) Enter the values in the calculator for this formula:
b=(nΣxy
–ΣxΣy)/(nΣx2- (Σx)2).
=(8*62750-521*960)/(8*33979-5212)
=4.7058…..
e) Now, calculate the value for "a" from the formula:
a= y-bar –b(x-bar)
=120-4.7058 *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*62750-521*960, STO, ALPHA, N.
2) Enter the
denominator and store it in variable D. 8*33979-5212 , 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,
L1 (or whatever list you want to sort); then press ENTER.
"Done" will be displayed
on the home screen,
indicating your data has been sorted. Note that you can also sort data in
descending order with SortD.
2. Finding Mean
(x-bar), ∑x, or ∑x2 , σ, Median, Q1, Q3
for Grouped or Ungrouped Data.
For Ungrouped Data:
a) After entering your data in the list as described in
item 1 of Section I, above, press STAT, and
cursor over to CALC, and press
ENTER. "1-Var Stats" will be pasted to the home screen.
b) Enter the list name you want to operate on by
pressing 2nd; then the list number, for example L1.
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 L1 , for example, and the corresponding data
values in L2.
For Grouped data:
a) Find the midpoints of each group and enter those
values in L1; then enter the corresponding frequencies
L2. Entering data in
a list is described in
item 1 of Section I, above.
b) Press STAT, cursor over to CALC, and press ENTER.
"1-Var Stats" will be pasted to the home screen.
c) Press 2nd, L1, 2nd, L2;
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 ∑x2
f is listed as ∑x2 .
3. Finding products
such as xy or (x-y):
a) Assume that your x-data is in L1 and your
y-data is in L2. Then obtain the product by pressing
2nd, L1; x (multiply
symbol), 2nd, L2, ENTER.
b) If you want the data stored in a list, L3
for example, before pressing ENTER in item a, press 2nd,
L1, STO, 2nd, L3.
Then press ENTER.
c) Obviously, x-y 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 L1.
b) If you want to display the data on the home
screen, press 2nd, L1; then the x2
symbol,
ENTER. The squared elements will be displayed.
c) If you want to store the squared data in a list, for
example L3, then press STAT, ENTER and place
the cursor over the list
name, L3 for example and press 2nd, L1, x2 ,
ENTER. If you want to replace
the original data with
the squared data in the same list, start the operation by placing the cursor
over list L1 .
d) If you want to multiply
corresponding elements of two lists and square each result; then your
expression should be like this:
(L1 * L2)2 .
5. Find x-x¯ (Sorry,
I have no symbol for the mean, so I displaced the bar.) from the data in
list L1.
a) Enter 2nd, L1, -, 2nd, LIST.
Note that" -" is a minus sign not a negative sign.
b) Cursor to MATH and press 3. You should
now have "L1-mean(" pasted to the home screen.
c) Press 2nd, L1, ENTER. The
result will be displayed on the home screen.
d) If you want to store the results in a list,
for example L3, then before ENTER in item "c" above, press
STO, 2nd, L3; then
ENTER
6. Finding (x-x¯ )2
a) Press (, 2nd, L1, -, 2nd, LIST.
b) Cursor to MATH and press 3. You
should now have "(L1-mean(" pasted to the home screen.
c) Press 2nd, L1,),),x2 .
The expression ((L1-mean(L1))2 should now be
displayed on the screen.
Press ENTER and the
results will be displayed on the home screen.
d) If you want to store the results in a
list, for example L3, before pressing ENTER in item "c"
above, press STO, 2nd, L3; then
ENTER.
7. Finding (Σx)2
and Σx2
Some computation formulas for the standard
deviation require (Σx)2 . To find that, do the following:
a) Enter your data in a list as described
at the beginning of this document. Press 2nd, QUIT to get
out of the list. Press (
to enter a parenthesis on the home screen.
b) Press 2nd, LIST, and cursor over to
MATH.
c) Press 5. "(sum(" should be entered
on the home screen.
d) Press 2nd, L1 or whatever
list your data is stored in.
e) Press ), ), x2 . You
now should have (sum(L1))2 on your home screen.
f) Press ENTER and the results will
be displayed on the screen.
g) Σx2 can be found by
using the "1-Var Stats" function under STATS, CALC, but you can also
find it by entering "sum
L12 "
8. Notice that you may
also do several other operations by pressing 2nd, STAT; then moving the cursor to
MATH and entering the list name that you wish to operate on.
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.
VI. Binomial
Distribution, pdf, cdf:
The binomial with n items, p probability, and
r trials can be calculated for a specific value using a calculator or
looked up in a table. Let's suppose,
however, that you have 12 items, each with a probability of 0.3, and you
are asked to find the probability that 6≥
r≤9. You can always look the values in the table, but it's much easier
if you use the little program below.
Prgrm:BINOMIAL
:INPUT "INPUT N", N
:INPUT "INPUT P", P
:INPUT "BEGIN R", K
:INPUT "END R", R
:{0}→L1
:R→dim (L1)
:For(V, K, R)
:N nCr V*P^V*(1-P)^(N-V)→L1(V+1)
:End
:sum L1→A
:Disp "P=", A
:Stop
To do the problem described above, just start the program, enter values for N
and P at the prompt. After that, enter
6 at the "BEGIN R" prompt and 9 at the "END R" prompt.
If you want just one value, put that value in for both BEGIN
and END. Of course, if you want Cdf through, say 5, just enter 0
(zero) for BEGIN and 5 for END. If by chance you
need the individual values, just look in list L1. You can get
there by pressing 2ND, LIST, ENTER.
VII.
Probability with the Normal Distribution:
Preliminaries:
Although students usually buy a TI-83 Plus, Casio CFX-9750, or newer calculator
when they take
statistics, some students may find themselves without funds to buy
a new calculator. For that reason, I am
including how to do probabilities with a normal distribution.
The method we will follow is to graph the normal curve using the
graphing function and integrate the curve within
limits to find the area, which can be interpreted as the
probability.
First we must set up the equation for the normal curve on the Y=
screen. We'll store it in the Yo= so that it will not
be in our way when we are graphing functions and doing plots.
Important: We will use M for mean and S for standard
deviation if the formula. We will store new values for these
variables when we have a new problem, but the equation will remain
the same.
Storing the equation:
a) Press Y= and move the cursor down to Yo=. Enter
the following: e-.5((x-M)/S)2 )/(S√(2π)) . If you are not
going
to use the equation
immediately, move the cursor to the equal sign opposite Yo and press ENTER to
deselect
that entry.
b) Now let's set the WINDOW. In general, set the window
Xmin at mean+ 5 standard deviations and Xmax at
mean + 5 standard deviations.
Set Ymin at 0 and Ymax at .4/S. That is .4 divided by the standard
deviation.
Now for examples:
Ex 1: Let's say we have a distribution with a mean of
70 and a standard deviation of 4. We want to find the
probability that a random selection is less
that 66.
a) Store 70, the mean, and the standard
deviation, 4, in the variable M and S respectively by entering 70 and
pressing,
STO, ALPHA, M, ALPHA, :,4, STO, ALPHA, S. You should now have this:
70→M:4→S. Press
ENTER and the
variable values will be stored. If you are ever in doubt about whether a
variable is
stored, just
press ALPHA the variable letter and ENTER. The stored value will be
displayed.
b) Press WINDOW and set enter 70-5*4 opposite
Xmin and 70+5*4 opposite Xmax. Set Xscl to 5 if you don't
like the thick
x-axis. Set Ymin at 0 and Ymax at .4/4 = .1
c) Press GRAPH and a standard curve should be
displayed.
d) Press 2nd, CALC, 7(∫f(x)dx), enter 50, the
number for Xmin, and press ENTER.
e) Move the cursor a few x-units to the right and
enter 66, ENTER. The area will be shaded and the value for
the area, 0.1586...
will be displayed on the screen.
Ex 2: Suppose that again we have a mean of 70 but a
standard deviation of 6. We want to find the
probability that a random selection will be
between 66 and 76.
a) Store the standard deviation, 6, from the home
screen in the variable S by pressing 6, STO, ALPHA, S.
You should now have this:
6→S. Press ENTER and the variable values will be stored. If you are
in doubt
about whether you have
the correct variable for M, just press ALPHA, M, ENTER and the
stored value will
be displayed.
b) Press WINDOW and enter 70-5*6 opposite Xmin and 70+5*6 opposite Xmax. Set Xscl to 5 if you don't
like the thick
x-axis. Set Ymin at 0 and Ymax at .4/7 = .07. Actually, Xmax and
Xmin only need be wide
enough to include
the values 66 and 76 for this particular problem.
c) Press GRAPH and a standard curve should be
displayed.
d) Press 2nd, CALC, 7(∫f(x)dx), enter 40, the
number for Xmin, and press ENTER.
e) Move the cursor a few x-units to the right and
enter 100, ENTER. The area will be shaded and the value for
the area, .5888...
will be displayed on the screen.
Ex 3: Finally let's suppose that again we have the same
problem as in Ex 2, but we want to find the probability
that a
random selection is greater than 76.
a) Assume the current mean, 70, and the standard
deviation, 6, are still stored and the WINDOW is set as in
Ex 2.
b) Press 2nd, CALC, 7(∫f(x)dx), enter 76, the
number for Xmin, and press ENTER.
c) Move the cursor a few x-units to the right and
enter 100, ENTER. The area will be shaded and the value for
the area, .15865..
will be displayed on the screen.
A FINAL WORD:
There are other things that could be done with the TI-83, but it is such an old
calculator that
I think it is more useful to spend my time with more
modern calculators. If you have question, I'll try to find
time to answer them.
Making it Better: I
would be grateful if you would report any errors or suggestions for improvements
to me. Just click "E-mail Webmaster," site the item number, and tell me
your suggested change.
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