Brief User Guide for TI-89 Titanium Statistics
This document covers
single- and two-variable statistics, scatter plot, regression analysis,
normalpdf, Student's Distribution, hypothesis
testing, statistics of two populations, and much more.
INDEX:
To facilitate lookup, the instructions are divided into
the following categories:
I. Data
Manipulation - Entering data, sorting data, using the APPS screen, friendly values from
graphs,
II. Single-Variable Statistics - Histogram by hand, simple histogram
with the calculator, frequency polygon
from
ungrouped data, cumulative frequency (Ogive) graph,
Relative frequency polygon and Cumulative relative
frequency Graphs, histogram using grouped data, frequency polygon using grouped
data, cumulative frequency
from grouped
data, relative frequency polygon and cumulative relative frequency graph from
grouped data,
percentile graph, box and whisker, box and whisker with outliers, box and
whisker by hand, discrete
probability
distribution, discrete probability distribution by hand.
III. Two Variable Statistics – scatter plot, regression analysis,
scatter plot and regression analysis on same
axis,
IV. Aids in doing statistics by hand
- Putting data in order, finding mean,
Σx, Σx^{2},
σ, median, Q1, Q3, finding
products such as xy and finding x-y, squaring elements of lists, finding x-xbar,
finding (x-xbar)², Σ(x-xbar)²,
finding Σ(x)^{2
}or Σx^{2 } .
V. Permutations, combinations,
factorials, random numbers, generating a random data set, selecting
numbers
randomly from a normal data set,
VI. Normal Distribution - Area under
a normal curve, Finding Z values, Graphing a curve, WINDOW
settings for graphing a curve, Probability Distribution Function using normalpdf(,
Graphing the
Normal
Distribution Using normalpdf(, normalcdf(, ZInterval,
VII. Other Distributions - TInterval,
invT Finding a t-value given α and df,
Chi-squared Distribution
VIII. Hypothesis testing - mean and z-test
(data), mean and z-test (statistics), mean and t-test (data),
mean and t-test (statistics).
IX.
Statistics of two Populations - confidence interval for two dependent
population, confidence interval for two
independent populations (Data and Stats),
IMPORTANT NOTICE: This is a "beta" version, which means that I have only
typed it out without actually
checking out the procedures on the calculator.
It should be reasonably accurate, but if you find errors, please
send me an e-mail. To do that, just click on "E-Mail Webmaster" in the
navigation bar and an e-mail form
will appear. I will do some more editing soon after the fall semester starts.
RELEASE DATE: Not
Released DATE LAST REVISED:
4/2/13
A beta version of the in printer friendly format appears here.
NOTE: See copy restrictions and printing
hints at the end of this document.
Preliminaries: We will be using lists a lot
in these procedures. You can assign your own variables to a list, but in
these procedures, I will be using list1 through list6. For
simplicity, I will call this the list tables. Many, perhaps most,
of these procedures will be done from the list screen, but some
operations, especially arithmetic operations, will be
done from the Home screen. There are several ways to paste list
variables either to the Home screen or to a dialog box.
Here are a couple:
a) To enter list1 press 2ND, ALPHA, l, i,
s, t, ALPHA, 1. That takes me about 12 seconds.
b) To enter list1, press 2ND, VAR-Link, l
(that's L not 1), ENTER. That takes me about 8 seconds. So guess
what,
I usually use
the last one. There are others which you can look into at your leisure.
I.
Data Manipulation
(NOTE: In
some instances you may want to clear a list or lists before you start entering
data. You
can overwrite data already in a list, but remember that
if the old list was longer than the new one,
you must delete the remaining old data an item at a
time. The easiest way to clear one of the tabular
lists, list1_{ }-list6, _{ }is to
place the cursor on the name above the list and press CLEAR; then ENTER.
1)
Entering Data:
a) To call up the
list tables press APPS, select the Stats/List icon and press ENTER. Either the
tables will
appear or you'll get a dialog box. If you get the dialog box just press
ENTER to go to the tables.
b) To enter data,
just place the cursor where you want to enter the data and press the correct
numbers. You don't have to erase old data if there is already data in the
list, but if the old list
is longer than the new list, you will need to delete the remaining old data
items. To delete an entry
just place the cursor over the data and press the backspace key, ← .
2) Putting Data in Order:
Go to the
lists following the instructions in Entering Data procedure above.
a) Place the
cursor in the list you want to sort and press
F3, 2, 1. On the dialog box that appears, the name of the
list you
selected should be entered. If it is not, enter it now by pressing 2ND, VAR-LINK,
l, (L, not 1); then cursor
down to the list name you
want and press ENTER
b) Move the
cursor down opposite Sort Order and select either "Ascending" or "Descending" as you
prefer by
pressing the right cursor key if you need to change the entry.
c) Press
ENTER and the list will be sorted. You may need to press ENTER again.
3) Using the APPS screen:
a) Press APPS to go to the screen that has
icons for all of the applications.
b) If the
application that you want is not on the first screen, press the first letter of the title of the icon you want
to
select. For example, if you want
to select Matrices press D, the first letter of the title
Data/Matrices. For Lists you would press S,
for "Stats/Lists."
c) Press 2ND, APPS
to toggle back and forth between the present screen and the previous APPS
screen. Note
that non-APPS screens may not be not recalled by this method.
d) Another way to
organize the APPS functions is to place them in folders. As for myself, I have
placed the
six APPS that I use most often in the MATH folder. That way
when I press APPS, I can then press
F1,2, 4 and my most-often-used APPS will immediately be
displayed on the screen. To organize the APPS, do
this:
1) Press APPS, F1, right cursor arrow, 3. A dialog box with a
list of APPS will appear on the screen. Scroll
up or down the list with the cursor arrows and press the
cursor right arrow to select one of the APPS.
Frankly, for most people, I think this is more trouble than it is worth.
4)
Friendly Values
on Graphs Using TRACE:
Many times when you use the TRACE function with a graph, you may get an x-value such as 2.784532. If
you
change the x-min and
x-max in the WINDOW function to be 7.9 or multiples of that number and
the y-min and y-max to multiples
of 3.8 or multiples, the displayed values will be
"friendlier." That is, they will be integers or numbers with one or two decimal
places. You can always set the values by hand, but the easiest method is to use
the ZoomDec function of ZOOM.
Just press F2; then 4, for ZoomDec.
If you want the more of the graph on the screen, press Zoom, set the cursor to
x=0 and y=0 and press
3 for ZoomOut. If
you want to reduce the covered area, press ZoomIn in a similar manner..
If you’re trying to find
the value at a specific point, say x=0, and the cursor does not fall on
the x-axis, you could use the zero function.
To do that, press F5, 2. That will set you up
for finding a zero. Select the lower bound (left); then the
upper
bound and press ENTER. Remember
that any time you want to get back to the decimal
window just press F2, 4.
II.
Single-Variable Statistics
1) Doing a
Frequency Distribution Histogram by Hand:
a) Use items
1, 2, and 3 in Section I above to enter and sort your data.
b) Find the
class width as follows:
(1) Let S represent the smallest data number (The first number in your
sorted list.), L be
the largest number (The largest number in the sorted list.), and C be the number
of
classes you've chosen. Find the class width, W, with the formula W =
(L-S)/C.
c) Determine
the limits of the classes by adding the class width to each successive class.
Don't forget that the lower class
limit is counted as part of the class width.
d) Determine
the number of data points in each class as follows:
(1) If your data is in list1, go to that list. Make sure your data is
sorted in ascending order;
then scroll down to the last number that falls within the upper limit of the first
class. At
the bottom of the list
(if you are using list1) your will see list1[#], where # is the number of data items
in
your first class.
(2) Scroll down to the last item of the second class and subtract the
number of items in the
first class from the number that appears in list1 (#). Continue this until
you come to the
end of the list. Note that if you also want cumulative frequency, just
write down the
numbers as you progress.
e)
Subtract 0.5 from each lower class limit of the first class to get the lower
boundary of the
first class. Add the class width to get successive boundaries.
f)
Alternatively, you could do the histogram described below and use the data
classes and
values from that histogram.
2) Doing a Histogram
with the TI-89 Titanium:
This procedure
describes how to do a simple histogram for which the user selects the class (bucket)
width and,
therefore, the number of classes.
First you need to get your data into lists.
a) First go
to the graphing screen by pressing ♦, F1 and deselect or clear any
functions or plots so
that they won't be
displayed with your graph.
Now, go to the lists and enter data as follows:
b) Press
[APPS], select the "Stats/List" icon and press [ENTER]. If the
Stats/List icon is not on the screen, you
can either search for it by using the cursor arrows or press S to display
the icons whose titles start
with an "S." Select the "Stats/Lists" icon and then press ENTER
c)
The list tables may be displayed, if not and your list tables are in the main
folder, press ENTER. If the lists
are not in the main folder, select the correct folder and press ENTER.
d) Enter the
data points in list1; then press F2,
1 to select "Plots Setup."
e) Press F1
and on the
dialog box that appears, select Histogram if it is not already displayed.
Do that by pressing
the right cursor arrow
and then pressing 4.
f) Move the
cursor down to the box opposite "x" and enter the name of the list that your data for the
histogram is in.
Do that with this sequence of
keystrokes: 2ND, VAR-LINK (the minus key), l (that's the letter L, not
the number 1)
and press ENTER if your data is in list1. If
the data is in another list, highlight that list and
press ENTER.
g) Move to the box
opposite "Hist. Bucket Width" and enter the number for your class width.
h) Make sure
that "Use Freq. and Categories" is set to NO and press ENTER.
i)
Press F5 to display the histogram. You may want to change the window
settings. Press
♦, F2 to change the
WINDOW settings.
At a minimum, I usually change y-min to 0, but that is a matter of personal choice. You
may want to use
F3 (Trace) to find the maximum y-value for your graph. You will also notice that the calculator
has set
the class boundaries. I usually set my own boundaries by setting x-min at 0.5 less than the smallest number,
but that
again is a matter of the practices
of your text or teacher. To get back to the histogram after setting the
WINDOW, press 2ND, APPS.
j) To display
the numbers for the boundaries of the classes and the number of
items in the class, press
F3 (Trace); then use the cursor to move across the
tops of the bars in the histogram and read the
various values for max, min, and n.
3) Constructing a Frequency Polygon from Ungrouped Data:
After graphing the histogram, you can use TRACE to get the data
for the frequency polygon and a cumulative
frequency graph if you wish.
a) Press TRACE and use the arrow to move across the histogram bars.
Record the values for x-min, x-max, and "n"
on a sheet of paper in tabular form.
b) Add one-half the class width to each x-min value
to get the midpoints and record those
values. Store these
values in a list, for example list2 if you have your histogram data in list1. Store the
corresponding values of "n"
in list3.
c) Press F2,
ENTER, F1, press the right cursor arrow; then 2 to select xyline.
d) Enter list2
opposite x and list3 opposite y. You can do this using either of the
methods I have previously described.
e) Press
ENTER and maybe ENTER again if the plot list screen does not appear.
f) Press F5
at the Plot Setup screen, and the graph will appear on the screen.
NOTE: Some teachers or texts prefer return-to-zero graphs. If your
course requires that, do the following after step b)
above:
A. Calculate a
midpoint of a new class preceding the first class
and another midpoint after the last class. These
values will be entered into
list2. To do that place the
cursor at the first item in list2, press 2ND, INS and replace
the zero that appears with your the first midpoint you calculated. Go to the bottom of the
list2 list and enter the
second value you calculated.
B. Now you want to enter
zero in list3 opposite each of these new midpoints. Place the cursor at
the top of list3 and press
2ND, INS. A zero will be added. Now cursor to the bottom of the list
and enter a zero opposite the last new midpoint
that you entered in list2.
C. Press ♦,
GRAPH and the graph will be displayed.
4. Constructing a Cumulative Frequency Chart (Ogive) Graph:
Suppose
that you have the frequencies in list3.
a) Enter the x-max
values that you recorded above in a list. For example, list4_{ }if you still have data in the
other lists.
If you want a return to zero graph you will need to calculate new maximum value
for the beginning and end of
end of this list.
b) Now, store the
cumulative frequency data in list5 _{ }as follows: Press
Home, 2ND, MATH, 3, 7 to paste cumSum(
to the Home screen.
Note that if you do not want a return-to-zero graph, you should delete the zeros
at the top and
bottom of this new list
c) Press 2ND,
VAR-LINK, l (L, not 1). Scroll to
list3 or whatever list you have the frequencies store in. Press ENTER
to paste that list to
the Home screen.
d) Close the
parentheses and press STO; then 2ND, VAR-LINK, l (L, not 1).
e) Scroll to list5,
or whatever list you want the cumulative frequencies stored in, and press ENTER.
The cumulative
frequencies will now be stored in list5.
f) Now you want to
graph list4 as" x" and list5 as "y."
g) Go to the list
screen by pressing 2ND, APPS, or by pressing APPS, selecting the Stats/Lists icon, and pressing ENTER.
h) From the list
screen, F2, ENTER, F1, press the right arrow; then 2 to select xyline.
i) Enter list4
opposite "x" and list5 opposite "y." You can either type the lists in the
dialog box or use VAR-LINK as
I have described previously.
j) Press
ENTER and maybe ENTER again if the plot list screen does not appear.
k) When the
plot screen appears, deselect all plots except the one of interest by pressing
F4.
l) Press F5 and the graph will appear on the screen.
NOTE: Some teachers
or texts prefer return-to-zero graphs. If your course requires that, and
you have not already
taken care of that in the
steps above, do
the following after step b) above:
A. Calculate an
x-max for a new class preceding the first class. This value will be entered into list4. To do that place
the cursor at the
first item in list4, press INS and replace the zero that appears with the
x-max you calculated.
B. Now you want to enter
zero in list5 opposite this new x-max value. Place the cursor at
the top of list5 and press
INS. A zero will be added.
C. Press ♦,
GRAPH and the graph will be displayed.
m) If the graph does not appear
on the screen, press ZOOM, 9
and the graph will appear on the screen.
5) Relative Frequency polygon and Cumulative Relative Frequency (Ogive)
Graphs:
Do
these exactly as in the frequency
polygon and cumulative frequency graph above except that after storing
the data (step
a) for the frequency polygon, do these
steps:
a) Press
Home, 2ND, VAR-LINK, move to list3, or wherever you have the
frequency data stored, and press
ENTER. The term list3 should now be displayed on the home screen.
b)
Press ÷, N, STO, 2ND, VAR-LINK, select list3 and press ENTER. This will replace the data in
list3 _{ }to with
relative frequency. Note that "N" is
a
number equal to the total number of data points that are in your sample.
c)
Press ENTER and the procedure will be executed. Now finish the procedure
as in the frequency polygon.
Notice that you will need to set y-max to a number smaller than 1.
6) Histogram Using Grouped Data:
a) First go to the graphing screen by pressing
♦, F1
and deselect or clear any functions or plots so that they will
not be displayed with your graph.
Now, go to the list and enter data as follows:
b) Press
[APPS], select the "Stats/List" icon and press[ENTER]. If the
Stats/List icon is not on the screen, you
can either search for it by using the cursor arrows or press S to display
the icons whose titles start with an
"S." Then press ENTER
c)
The list tables may be displayed, if not and your list tables are in the main
folder, press ENTER. If the lists
are not in the main folder, select the correct folder and press ENTER.
d)
Enter the
midpoints of the classes into list1 and the corresponding frequencies into
list2, or whatever lists
you choose. Press F2,
1 to select "Plots Setup."
e) On the
dialog box that appears, select Histogram if it is not already displayed.
Do that by pressing the right
cursor arrow and pressing 4.
f) Move the
cursor down to the box opposite "x" and enter the name of the list that your data for the
histogram is in.
Do that with this sequence of
keystrokes: 2ND, Var-Link (the minus key), l (that's the letter L, not
the number 1)
and press ENTER if your data is in list1. If
the data is in another list, highlight that list and
press ENTER.
g) Move to the box
opposite "Hist. Bucket Width" and enter the number for your class width.
h) Make sure
that "Use Freq. and Categories" is set to YES, and enter the list
where the frequencies are stored
opposite Freq. Press ENTER, and maybe ENTER again if the Plot Screen does not appear.
i)
Press F5 to display the histogram. You may want to change the window
settings. At a minimum, I usually
change y-min to 0, but that is a matter of personal choice. You may want to use
F3 (Trace) to find the maximum
y-value for your graph. You will also notice that the calculator has set
the class boundaries. I usually set my
own boundaries by setting x-min at 0.5 less than the smallest number, but that
again is a matter of the practices
of your text or teacher.
j) To display
the numbers for the boundaries of the classes and the number of
items in the class, press
F3 (Trace); then use the cursor to move across the
tops of the bars in the histogram and read the
various numbers.
7) Frequency Polygon Using Grouped Data:
Do this exactly like the histogram, except select the
2 (xyline) rather than Histogram. If you've already done the
histogram,
just change "Plot Type" and press ♦, F3.
8) Cumulative Frequency (Ogive) Graph from Grouped Date:
a) Enter
the upper class limits in a list, for example, list3 _{ }if you have data in the first two lists.
b) Enter the
frequencies in list2 if it is not already there. Now, do the following:
A) From the list screen highlight the title of list4, press 2ND, MATH, 3, 7. The expression cumSum( will
be posted to
the
area at the bottom of the list screen.
B) With the cursor after the parenthesis, press 2ND,
VAR-LINK, L, cursor to list 2 and press ENTER. Close the _{
}
parentheses and press_{,}STO, 2ND, VAR-LINK, L, cursor to list4 and press
ENTER_{ }. You will now have
cumSum(list2) pasted at
the bottom of the screen. Press ENTER. The cumulative frequencies will now be
in list4. Now, you want to plot the upper class limits as "x" and the
cumulative frequencies as "y."
c) Press F2,
1 to select "Plots Setup." Select the plot number you want with
the cursor arrows.
d) From the list
screen, press F1 (Define) and change the "Plot Type" to xyline
on the dialog box that appears.
Do that by pressing the right cursor arrow and pressing 2.
f) Move the
cursor down to the box opposite "x" and enter the name of the list that your
data for the midpoints is in.
Do that with this sequence of
keystrokes: 2ND, VAR-LINK (the minus key), l (that's the letter L, not
the number 1)
and press ENTER if your data is in list1. If
the data is in another list, highlight that list and
press ENTER.
g) Move to the box
opposite "Hist. Bucket Width" and enter the number for your class width.
h) Make sure
that "Use Freq. and Categories" is set to YES, and enter the list
where the frequencies are stored
opposite y. Press ENTER, and maybe ENTER again if the Plot Screen does not appear.
i)
Press F5 to display the frequency polygon. You may want to change the window
settings. At a minimum,
I usually change y-min to 0, but that is a matter of personal choice.
9) Relative Frequency and Cumulative Relative Frequency Graphs for
Grouped Data:
Do these
exactly as in the frequency polygon
and cumulative frequency graph above except that after storing
the data for the frequency polygon do this step:
a) Press 2ND,
VAR-LINK, move the cursor to list4 and press ENTER.
b)
Close the parentheses and press 2ND, VAR-LINK, move the cursor to list4 and
press ENTER.
c)
Press ÷, N, STO, 2ND, VAR-LINK, move the cursor to list4. Note that "N" is
the number for the quantity of data,
points not the letter.)
d)
Press ENTER and you should have list4/N→list4
pasted to the home screen.
e)
Press ENTER and the data in list4_{ } will be converted to relative frequency. This assumes
that the frequency
data is stored in list4_{ }
.
f) Continue
with the plotting as in the cumulative frequency graph above.
10) Percentile Graphs:
This
graph is fairly similar to the Ogive graph. We will do this in two groups
of steps: Preparing data
and plotting
data.
Preparing
Data:
a)
Enter upper boundaries in list1 and the corresponding frequencies in list2.
If you want the graph to start
at zero, enter the first lower boundary with zero for the frequency.
b) From
the Home screen, press
2ND, MATH, 3, 7. The term cumSum( will
be pasted to the Home screen.
c) Press 2ND,
VAR-LINK, l (L not 1), cursor to list2 and press ENTER. Press ), ÷ . You now should have
cumSum(list2)/ on the
home screen.
d) Press
2ND, MATH, 3, 6. You should now have
cumSum(list2)/Sum( on the Home screen.
e) Press
2ND, VAR-LINK, L, cursor to list2, and press ENTER. Close the
parentheses. You now should have
cumSum(list2)/Sum(list2)
on the home screen.
f) Press x
(the multiply symbol), 100, STO, 2ND, VAR-LINK, l, cursor to list3, and press
ENTER. You now should
have
cumSum(list2)/Sum(list2) *100→list3
pasted to the home screen.
g) Press
ENTER and the data will be stored in list3 .
Plotting the Data:
a)
Now you want to graph list1 as x and list2 as y, so press F2, ENTER, F1, 2
to select xyline.
b) Enter list1
opposite x and list2 opposite y. You can do this using either of the
methods I have previously
described.
c) Press
ENTER and maybe ENTER again if the plot list screen does not appear.
d) When the
plot screen appears, deselect all plots except the one of interest by pressing
F4.
e) Press F5 and the graph will appear on the screen.
f)
You can find the exact percentiles of the boundaries by using Trace (F3), and approximate
percentiles of
other x-values by using the cursor.
11) Box and Whisker Plot
a) First go to the graphing screen by pressing the
♦, F1.
.
Deselect or clear any Y= functions or plots so that
they won't be entered on your graph.
If you choose, clear the list as described at the beginning
of this document.
b) Press APPS, S,
highlight the Stats/Data icon and press ENTER
to go to the list tables.
c)
Enter your numbers in list1. (Or whatever list you choose.)
d) Press F2, 1, F1, press
the right cursor arrow, press 3 for Box Plot.
e) Opposite "x" enter
list1 or whatever list your data is in. To do that press 2ND, VAR-LINK, l
(that's L, not 1).
If your data is in list1 press ENTER, otherwise scroll to the list your data is
in and press ENTER.
g) Make sure "Use Freq.
and Categories" is NO; press ENTER and the Plot Screen should appear. If
not, press
ENTER again.
h) Press F5 and the box-and-whisker plot will
appear on the screen.
i) To find the numbers for the limits of the quartiles, press
F3 (Trace); then use the cursor to move
across
the diagram and obtain the values for quartiles or the beginning and ending
values.
12) Box and Whisker Plot with Outliers:
You may have one or
two outliers (numbers much larger than the rest) and you may not want to include
those
data plots in your box and
whisker graph because they will distort the graph. You can do a modified
box plot
as follows:
a)
Do the plot exactly as in the Box and Whisker above except in step d), select 5
(Mod Box Plot) instead of
Box Plot.
b) This will not include the outliers in the last whisker, but will plot them as
separate points
after
the end of the last whisker.
13) Box and Whisker Plot by Hand
You can save yourself considerable
calculation if you use the calculator to find Q_{1}, Median, and Q_{3}
when doing a box-and-whisker plot by hand.
To find those values do the following:
a) Press APPS, S and select the
Stats/List icon; then press ENTER. This will take you to the tabulated
list
screen.
Enter your data in a list, for example, list1.
b) Press F4, 1. This will take
you to the 1-Var Stats dialog box.
c) Opposite "List" enter the list
number where your data is stored, for example list1. You can either type
in
the name or
press 2ND, VAR-LINK, l (L, not 1), and press enter when the correct list number
is highlighted.
d) Type 1 in the box opposite Freq.
and press ENTER. If the statistic do not appear after a few seconds, press ENTER again.
e) Cursor down and you will find
Q _{1} , Q_{3} , and Med, Max and other statistics.
f) Draw the box plot using your usual
method and using these numbers.
14) Discrete Probability Distribution
Let's take a simple example to
demonstrate this: Suppose a word is flashed on a screen several
times while people are trying to
recognize the word. The list below indicates what percentage of the
group required a given number of flashes to
recognize the word.
No. of Flashes 1 2
3 4 5
Percent
27 31 18
9 15
P(x) .27 .31 .18
.09 .15
In summary, the method is to
enter the number of flashes into list1 and the corresponding P(x)
values into list2 as
the frequency. The details are as follows:
a) Enter the number
of flashes in list1 and the corresponding P(x) values in list2_{
}opposite the
number of flashes. (How to enter data in a list is covered at the
beginning of this document.)
b) From the tabulated list
screen, press F4, 1. This will take you to the 1-Var Stats dialog box.
c) Opposite "List" enter the
list number where your data is stored, for example list1. You can either
type in
the name or
press 2ND, VAR-LINK, l (L, not 1), and press enter when the correct list number
is highlighted.
d) Enter list2 in the box
opposite Freq. and press ENTER. If the statistic do not appear, press
ENTER again.
e) If you need the
variance, merely re-enter the value for the standard deviation, 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 x-values should be stored in list1
and
the p(x)
values in list2.)
The mean can be obtained by the following formula: mean = Σxp(x).
To obtain the
individual values and store them in list3, do the following after
storing the data :
a)
From the Home screen press 2ND, VAR-LINK, S, highlight list1 and press ENTER.
b)
Press the multiply sign, x; then press 2ND, VAR-LINK, S, highlight list2 and
press ENTER. You
should now have list1*list2 displayed on the Home screen.
c)
Press 2ND, STO, 2ND, VAR-LINK, S, highlight list3 and press ENTER.. You
should now have
list1*list2→list3 pasted to the home screen.
d)
Press ENTER and you will have the individual values stored in list list3_{ }
and displayed on the
home screen. This is the products xp(x).
e)
To get the sum of these values, do this.
(1)
Press 2ND, MATH, 3, 6. The expression sum( will be
pasted to the Home screen.
(2) Press 2ND, VAR-LINK , l (L, not 1); select list3 and press ENTER.
(3) Close the parentheses and press ENTER
You can
obtain the variance and standard deviation by first solving for the variance
using
the formula: Σx^{2} P(x) -
µ^{2} where µ is the mean obtained as above. To obtain the
individual
values
of the first term, x^{2} P(x).
and store them in list L_{4}, do the following:
a) Press 2ND, VAR-LINK, l (L, not1), select list1 or whatever the
x-values are in and press ENTER.
b) Press ^, 2, x (multiply sign), 2ND, VAR-LINK, l (L, not1), highlight
list2 and press ENTER.
c) Press STO,
2ND, VAR-LINK, cursor to list3 and press ENTER. You will have list1^{2}*list2→list3_{
} _{ }pasted to the
Home screen.
d) Press ENTER and the individual values will be entered in list3_{
}and pasted to the home
screen.
e) To get the sum of these values do the following:
(1)
Press 2ND, MATH, 3, 6. The expression sum( will be
pasted to the home screen.
(2) Press 2ND, VAR-LINK, l (L, not1), select list 3, close the
parentheses and press ENTER.
f) Now
subtract the value for µ^{2} from the last value obtained and that will
be the variance.
g) To obtain the standard deviation
from the variance given in the 1-Var Stats, do the following:
(1) From the home screen, press 2ND, √ , the
multiply key; then enter the number for the variance,
close the parentheses, and press ENTER.
NOTE:
Obviously, if you only want to obtain the values for the these three
parameters, you can
use this
method, but it is much easier to use method 15 above. Just for the
information of the reader, the
total
expression for the variance using this method would the this: sum(list1^{2}*list2)
- (sum(list1 *list2))^{2} .
III. Two-variable Statistics
1)
Scatter Plot
First you need to get your data into lists.
a) Go to the graphing screen by pressing the
♦,
Y= and
deselecting any functions so that
they won't be entered on your graph.
If you want to clear the lists before entering data, see the
note at
the beginning of this document.
b) Press
APPS, S, highlight the Stats/List icon and press ENTER. If the
tabulated lists do not appear, press
ENTER again.
c) Enter the data-point numbers ( the
x-values) in list1 and the
corresponding values (y-
values) in list2. (If your data is not in order you can sort in order by
placing the cursor in the list
that
you want to sort and pressing F3, 2, 1. The list where the cursor is
located should be in the
box
opposite "List." Move the cursor to "Sort Order" and select either
ascending or descending
as you
choose. Press ENTER to sort the list. BE CAREFUL! If
your data in list2_{ }is not in ascending order
when correlated
to list1_{, }
then your data in list1 and list2
will not be correlated correctly after sorting.
d) Press F2, ENTER to go to the Plot Setup
screen.
e) Using the right cursor arrow, change the
Plot Type to Scatter if it's not already there; then enter list1 in the
box opposite "x"
and list2 opposite "y."
f) Press ENTER; then at the "Plot Setup"
screen that appears, press F5 and the scatter points will be plotted.
2) Plotting x-y line Graph:
Do that the same
as the scatter plot in item 1 above except that when you select the "Plot Type,"
choose
2: xyline rather than scatter.
3) Regression Analysis
First
you need to get your data in lists. You can do that from the home screen, but if you have any
significant amount
of data, it's much easier to enter it into list tables. See the note at
the beginning of
this document for instructions on clearing lists if you want to
clear your lists before data entry.
Here's how to enter data:
a)
Press APPS, S, highlight the Stats/List icon and press ENTER.
If the tabulated lists do not appear, press
ENTER again.
b) Enter the numbers for the independent variable,
x-values, in list1 and the corresponding values in list2.
c) From the list screen, press F4, 3
and choose the regression equation of your choice, for example 2, for
LinReg (ax+b).
Press the number for your selection.
d) On the dialog box that appears, enter
list1 opposite "x" and list2 opposite "y." Enter list1 by pressing
2ND, VAR-LINK, l (L, not 1),
highlight list1 and press ENTER. Move the cursor down opposite "y" and
repeat the
steps to enter list2 there.
e) If you are planning to graph the equation, the
box opposite "Store RegEq to"
should have y1(x)
in it. If not, press the right arrow and then ENTER. Enter 1 for Freq. if in is not already
there and
press ENTER.
f) The values for a, b, r and r2 will
appear on a dialog box.
Graphing
the Regression Equation:
a) If you want to graph the
regression equation, press
♦, Y=. That will take you to the Y= screen.
b) Clear or deselect all of the entries
except Y1 and check y1by pressing F4 if it's not already checked.
c) Press
♦, GRPH
and the graph will be
displayed. You may need to adjust the window to see the graph.
4) Plotting a graph with the scatter plot and the
regression equation on the same axis.
First you need to get your data in lists and do the regression graph as described above in item 3.
Make sure you have told the calculator where to store the regression equation in step e)
above. Now, you want
to put the scatter plot on the screen with the graph. To do this:
Scatter Plot
We will assume that your data are already in
list1 and list2.
d) Press F2, ENTER to go to the Plot Setup
screen.
e) Using the right cursor arrow, change the
Plot Type to Scatter if it's not already there; then enter list1 in the
box opposite "x"
and list2 opposite "y."
f) Press ENTER; then at the "Plot Setup"
screen that appears, press F5 and the scatter points along with
the graph of the
regression equation will be displayed.
IV. Aids in doing statistics by hand.
General: Often in book problems in school you'll need to do a lot of
calculations by hand. These
techniques will save you a lot of arithmetic.
1. Putting Data in
Order:
a) Place the
cursor in the list you want to sort and press
F3, 2, 1.
b) Move the
cursor down opposite Sort Order and select either ascending or descending as you
prefer by
pressing the right cursor key if you need to change the entry.
c) Press
ENTER and the list will be sorted.
2. Finding Mean
(x-bar), ∑x, or ∑x^{2} , σ, Median, Q_{1}, Q_{3}
for Grouped or Ungrouped Data.
For Ungrouped Data:
a) Press APPS, S and
select the Stats/List icon; then press ENTER. This will take you to the
tabulated list
screen where you can enter your data
b)
The list tables may be displayed, if not and your list tables are in the main
folder, press ENTER. If the lists
are not in the main folder, select the correct folder and press ENTER.
c )
Enter your data into list1 or whatever list you choose.
d) Press F4, 1
for 1-Var Stats.
e) On the dialog
box that appear, enter list1 in the box opposite "Lists," and enter 1 opposite
the "Freq."
box.
f) Press ENTER and
various statistics will be displayed.
NOTE: You can also find these values for
discrete random variable statistics by entering the values
of the variable in list, for example, and the probabilities corresponding data
values in list2.
For Grouped data:
a) Press APPS, S
and select the Stats/List icon; then press ENTER. This will take you to
the tabulated list
screen where you can enter your data
b)
The list tables may be displayed, if not and your list tables are in the main
folder, press ENTER. If the lists
are not in the main folder, select the correct folder and press ENTER.
c )
Enter your x-values into list1 and the frequencies in list2, or
whatever lists you choose.
d) Press F4, 1
for 1-Var Stats.
e) On the dialog
box that appear, enter list1 in the box opposite "Lists," and enter list2
opposite the "Freq."
box.
f) Press ENTER and
various statistics will be displayed.
3. Finding products
such as xy, (x-y):
a) Assume that your x-data is in list1_{ }and your
y-data is in list2.
b) Press Home; then obtain the product by pressing
2ND, VAR-LINK, l (L, not1), scroll to list1 if not already
there, enter x (multiply
symbol), 2ND, VAR-LINK, l (L, not1) ENTER.
c) Press ENTER and the result will be displayed
on the home screen.
d) If you prefer to have the data stored in a
list, list3_{
}for example, before pressing ENTER in item c, press STO, _{
}
2ND, VAR-LINK, l (L, not1), cursor to list3 and press ENTER. _{
}
e)
Then press ENTER and the results will be displayed.
f) Obviously, x-y (x minus y) can be obtained by merely
substituting the subtraction symbol for the
multiplication symbol in
step a) above.
4. Squaring operations
such as elements of lists.
a) To square the elements of a data set, first
enter the data in a list, for example list1.
b) Press 2ND, VAR-LINK, l (L, not 1), select the
list you want and press ENTER.
c) Press ^, 2, ENTER. The squared elements will be displayed.
d) If you want to store the squared data in a list, for
example list3, then before pressing ENTER in
item c above, press STO,
2ND, VAR-LINK, l (L, not 1), cursor to the list you want and press ENTER.
Press ENTER again and the
data will be stored in list3.
e) If you want to multiply corresponding elements
and square each product, your entries should result
in the following:
(list1*list2)^2.
5. Find x-x¯ (Sorry,
I have no symbol for the mean, so I displaced the bar.) from the data in
a list.
a) Enter the data in list1.
b) From the Home screen, press 2ND, VAR-LINK, l
(L, not1), cursor to list1 and press ENTER; then press the
minus sign not a negative sign.
c) Press 2ND, MATH, 6,4. You should
now have "list1-mean(" pasted to the home screen.
d) Press 2ND, VAR-LINK, l (L, not 1), highlight
list1 and press ENTER.
e) Close the parentheses and press ENTER. The
result will be displayed on the home screen.
f) If you want to store the results in a list,
for example list3, then before ENTER in item "e" above, press
STO, 2ND, VAR-LINK, l (L,
not1), cursor to list3 and press ENTER.
g) If you did not have Exact/Approximate in the
MODE set to Approximate, you will have some terrible looking
fractions. To avoid that,
press MODE, scroll down to Exact/Approximate, press the right cursor arrow, and
then press 3.
6. Finding (x-x¯ )^{2
} and Σ(x-x¯)^{2}
a) Enter the data in list1.
b) From the Home screen, press (, 2ND, VAR-LINK,
l (L, not1), cursor to list1 and press ENTER; then press the
minus sign not a negative sign.
c) Press 2ND, MATH, 6,4. You should
now have "list1-mean(" pasted to the home screen.
d) Press 2ND, VAR-LINK, l (L, not 1), highlight
list1 and press ENTER.
e) Close the parentheses; then press ^, 2.
You should have the expression (list1-mean(list2))^2 displayed
on the Home screen.
f) Press ENTER. The
result will be displayed on the home screen.
d) If you want to store the results in a list,
for example list3, then before ENTER in item "f" above, press
STO, 2ND, VAR-LINK, l (L,
not1), cursor to list3 and press ENTER.
f) If you did not have Exact/Approximate in the
MODE set to Approximate, you could have some terrible looking
fractions. To avoid that,
press MODE, scroll down to Exact/Approximate, press the right cursor arrow, and
them press 3.
g) The expression
Σ(x-x¯)^{2} is evaluated in the 1-Var Stats.
To get that do the following:
(1) Press APPS, S and select the Stats/List icon; then press
ENTER. This will take you to the tabulated list
screen.
Enter your data in a list, for example, list1.
(2) Press F4, 1. This will take
you to the 1-Var Stats dialog box.
(3) Opposite "List" enter the list
number where your data is stored, for example list1. You can either type
in
the name or press 2ND, VAR-LINK, l (L, not 1), and press ENTER when the correct list number
is
highlighted.
(4) Type 1 in the box opposite Freq.
and press ENTER. If the statistic do not appear, press ENTER again.
(5) Cursor down and you will find Σ(x-x¯)^{2}
listed.
7. Finding (Σx)^{2}
and 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.
b) Press Home, (, 2ND, MATH, 6.
The expression "(sum(" should be entered on the home screen.
c) Press 5.
d) Press 2ND, VAR-LINK, l (L,not 1), cursor
to list1 (or whatever list you choose, and press ENTER.
e) Press ), ), ^,2 . You now should
have (sum(list2))^{2} on your home screen.
f) Press ENTER and the results will
be displayed on the screen.
g) Σx^{2} can be found by
using the "1-Var Stats" function by pressing F4, 1, but you can also
find it by entering "sum
list1^{2} "
8.
Finding the Standard Deviation by hand using the above exercises:
The standard
deviation computation formula is as follows:
s =
√(Σx² -(Σx)²)/n)/(n-1)
So, one can use
Σx^{2}_{ }and
_{ }(Σx)^{2} calculated above to calculate the sample
standard deviation.
9. Notice that you may
also do several other operations by doing the "1-Var Stats" which is covered in
another section
of this document.
V. Permutations, combinations, factorials, random
numbers:
1. Finding Permutations.
a) Suppose we want the
permutations (arrangements) of 8 things 3 at a time. Go to the home
screen.
b) Press 2ND, MATH, 7, 2. You will have nPr( pasted to the
Home screen.
c) Enter 8, 3. ), so that you have nPr(8, 3), and press
ENTER. You will get 336.
2.
Finding Combinations:.
a) Suppose we want the
combinations (groups) of 8 things 3 at a time. Go to the home screen.
b) Press 2ND, MATH, 7, 3. You will have nCr( pasted to
the Home screen.
c) Enter 8, 3. ), so that you have nCr(8, 3), and press
ENTER. You will get 56.
3. Finding Factorials.
a) Suppose we want 5 factorial (5!). Go to
the home screen.
b) Enter 5, ♦; then press the divide key. You will have 5! pasted to the screen.
c) Press ENTER and the
answer, 120, will be displayed.
4. Generating a random
number between 1 and n:
a) Suppose we want to generate a random
number between 1 and 15. Go to the Home Screen.
b) Press 2ND, MATH, 7, 4 and you will have
rand( pasted to the home screen.
c) Enter 15, ) and press ENTER. A
random number between 1 and 15 will be displayed.
5. Randomly
generated data sets:
Sometimes problems use a randomly generated set
of data. Suppose we want to generate 10
random numbers between 1 and 50 and store them in
list1. The proper syntax is randint(lower,
upper, how many). That can be obtained
as follows starting from the Stats/List.
a) Clear list1 by highlighting
the title, list1, press Clear, ENTER. Highlight the title again while
you do the
additional steps below.
b) Press F4, 4, 5. The expression randInt(
will be displayed at the bottom of the screen after list1=.
c) Enter 1, 50, 10, ), so that your screen
displays randInt(1,50,10).
e) Press ENTER and the numbers generated
will will be stored in list list1.
6. Normally distributed
data set:
Suppose you want a set of 10 numbers from a data
set with mean 50 and standard deviation 10.
The proper syntax is randNorm(mean,std. dev.,
quantity) That can be obtained
as follows starting from the
table of lists:
a) Clear list1 by placing the highlighting
the title, list1, press Clear, ENTER. Highlight the title again while
you do the
additional steps below.
b) Press F4, 4, 6. The expression .randNorm(
will be displayed at the bottom of the screen after list1=.
c) Enter 50, 10, 10, ), so that your screen
displays .randNorm(50, 10,10).
d) Press ENTER and the numbers generated
will will be stored in list list1.
VI.
Normal Distribution:
Note:
In this section, a general method will be
outlined; then a specific example will be worked. The
same
problem will be used in several of the examples.
General, normalcdf(: This function returns the value of the area between two
values of the random variable
"x." This can be interpreted as the probability that a randomly selected variable will fall
within those two
values of "x," or as a percentage of the x-values that will lie within that range. The syntax for
this function is
normalcdf( lower bound, upper bound, μ, σ. If the mean and standard deviation are not given, then the
calculation defaults to the standard normal curve with a mean of 1 and a standard deviation of 0. I use the
values -1E9 and
1E9 for left or right tails. The E in obtained by pressing 2nd, EE.
This can be used to solve
such problems as the following: P(x<90), P(x>100),
or P(90<x<120).
If 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) From the table of lists,
press F5, 4 and a dialog box for "normalcdf" will appear.
b) Enter the number for the
lower value, Upper Value, μ, and σ in that order.
c) Press ENTER, and the value of
the area between the two points will be displayed. Notice that
you do
not explicitly convert the points to z-values as in the hand method.
Ex. 1: Assume a
normal distribution of values for which the mean is 70 and the std. dev. is 4.5.
Find the probability that a
value is between 65 and 80, inclusive.
a) Complete item a)
above.
b) Enter
numbers 65 for Lower Value, 80 for Upper Value, 70 for mean, and 4.5 for σ.
c) Press ENTER (
you may need to press ENTER again) and
you'll get 0.85361 which is, of course,
85.361 percent.
2.
normalcdf(: Area under a curve to the left of a point with μ (mean) and σ (std.
dev.) given.
Ex. 2: In the
above problem, determine the probability that the value is less than 62.
a) Complete
item a) in the general method above.
b)
Enter numbers -1E9, for Lower Value, 62 for Upper Value, 70 for mean, and 4.5
for σ. Notice that
the "-" is a negative sign, not a minus sign. Enter "E" by pressing the
EE button.
c) Press ENTER (
you may need to press ENTER again) and you'll get 0.03772 which is, of course,
3.772 percent.
3.
normalcdf(: Area under a curve to the right of a point with μ (mean) and σ (std. dev.)
given.
Ex. 3: In the
above problem, determine the probability that a value is greater than or equal to
75.
a) Complete
item a) in the general method above.
b)
Enter numbers 75 for Lower Value, 1E9 for Upper Value, 70 for mean, and 4.5 for
σ. Notice that
the "-" is a negative sign, not a minus sign. Enter "E" by pressing the
EE button.
c) Press ENTER (
you may need to press ENTER again) and you'll get 0.13326 which is, of course,
13.326 per percent.
4. ShadeNorm(: Displaying a graph of
the area under the normal curve.
General:
This function draws the normal density function specified by
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
pressing ♦, F1 and either pressing
F4 to disable each function or press clear to erase the function.
b) Press 2ND,
APPS to go to the list tables. If this doesn't work, press APPS, highlight
the Stats/List
icon and press ENTER to go to the list tables.
c) Press F5, 1,1 and the Shade Normal dialog box will appear.
home screen.
d) Enter your
parameters, for example -1E9 for Lower Value, 62 for Lower Value, 70 for
µ, and 4.5
for σ.
e) Cursor
down to Auto Scale and change it to YES by pressing the right cursor arrow and
selecting YES.
f) Press
ENTER, and the graph may be visible on the screen. You may want to
reset the WINDOW, but
with the setting on Auto-scale, the graph usually looks satisfactory.
Ex 1:
Draw the graph of example 2 above.
a) From the Lists screen, press F5, 1,1
b) Enter your paramenters in the dialog box, for example -1E9 for Lower
Value, 62 for Lower Value, 70 for µ,
and 4.5 for σ.
c) Cursor down to Auto Scale and change it to YES by pressing the right
cursor arrow and selecting YES.
d) Press ENTER and a reasonable looking graph should appear on the screen.
5. invNorm(: Inverse Probability Calculation:
Find the number x, in a normal distribution such that a number is less than x
with a given
probability. The default is
the standard curve with mean=0
and standard deviation. is 1.
Ex. 1:
In Ex. 1 immediately above, find the number x, such that a randomly selected number will be
below
that
number with a 90% probability.
a)
Press F5, 2, 1 and the Inverse Normal dialog box will appear.
b)
Enter numbers .9 for Area, 70 for µ,
and 4.5 for σ.
c)
Press ENTER and your answer will be 75.766.
Ex.
2: Given a normal distribution with a mean of 100 and standard
deviation of 20. Find a value X_{o} such
that the
probability of the given x-value is below X_{o} is .6523. That is P(X<X_{o})
= .6523.
a) Press F5, 2, 1 and the Inverse Normal dialog box will appear.
b) Enter numbers .6523 for Area, 100 for
µ, and 20
for σ.
c) Press ENTER, and perhaps ENTER again, and your answer will be 107.83.
Ex. 3: What is the lowest score possible to be in the upper 10% of
the class if the mean is 70 and the
standard deviation is 12?
a) Press F5, 2, 1 and the Inverse Normal dialog box will appear.
b) Enter numbers .9, (1-.1)for Area, 70 for
µ, and 12
for σ.
c) Press ENTER, and perhaps ENTER again, and your answer will be
be 85.38 or 86 rounded off.
6.
ShadeNorm(: Graphing (shading) the Probability area:
Ex. 1: Obviously
if you wanted to graph the example immediately above, you could use the
ShadeNorm(
using the lower bound of -1E9 and the upper bound of 75.766. You would do
that
as follows:
a) Press F5, 1, 1 and the
Shade Normal dialog box will appear.
b)
Enter numbers -1E9 for Lower Bound, 75.766 for Upper Bound, 70 for
µ, and 12
for σ. Be sure
that Auto-Scale is set for YES. Do that by pressing the right cursor arrow
and selecting YES.
c) Press ENTER and a reasonable looking graph should appear on the screen.
Note that if you wanted to shade the region where the probability would be
greater than 90%,
you would choose 75.766 for the lower boundary and 1E9 as the
upper bound.
Ex. 2:
Suppose you wanted to graph a distribution and shade the area between the points 40 and 54,
with a mean of 46
and a std. dev.
of 8.5
a) Press F5, 1, 1 and the
Shade Normal dialog box will appear.
b)
Enter numbers 40 for Lower Bound, 54 for Upper Bound, 46 for
µ, and 8.5
for σ. Be sure
that Auto-scale is set for YES. Do that by pressing the right cursor arrow
and selecting YES.
c) Press ENTER and a reasonable looking graph should appear on the screen.
Note that if you wanted to shade the region where the probability would be
greater than 90%,
you would choose 75.766 for the lower boundary and 1E9 as the
upper bound.
7. normalpdf(: Probability Distribution Function using normalpdf( :
General: This function is used to find the fraction, and therefore
also the percentage, of the
distribution that corresponds to a particular value of x. The syntax of
this function is
normalpdf(X, μ, σ).
A)
Finding the Percentage of a Single Value:
Ex. 1: Suppose that the mean of a certain distribution is 60 and the
standard deviation is 12.
What percentage of the population will have the value 50?
a) Press F5, 3 and the Normal Pdf dialog box will appear.
b) Enter numbers 50 "X value" , 60 for
µ, and 12
for σ.
c) Press ENTER and your answer should be .023493 which is about 2.3
percent.
B)
Graphing the distribution:
Ex. 1: Suppose that
the mean of a certain distribution is 60 and the standard deviation is 12.
Investigate percentages for several x-values.
a) First press WINDOW and set Xmin 12 (mean minus 4 std. dev.). Set Xmax at the same
number of units above the mean, i.e., 108. Set Ymin=0 and Ymax = .05
b) Press ♦, F1 to go to the graphing screen. Select position y1.
c) Press CATALOG, F3, n, select normpdf( and press ENTER.
the Y1= position.
c) Enter data so that the entry after Y1= looks line this:
TIStat.normalpdf(X, 60,12). Be sure to
close the parentheses.
d) Press ENTER to transfer the entry to y1; then press F2, ALPHA, a, to select ZoomFit.
The curve should
appear on the
screen.
e) Press F3 and you can move along the curve and read the values for
different x-values. If you want a
specific value, perhaps to get rid of the x-value
decimals, just enter that number and press ENTER.
8. ZInterval: This gives the range within which the population mean is expected to fall
with a desired
confidence level. The sample size should be > 30 if the
population standard devation is not
known.
Ex. 1: Suppose we have a sample of 90 with sample mean x¯ =
15.58 and s = 4.61. What is the 95%
confidence level interval?
a) From the lists screen press 2ND, F2, 1 and the ZInterval dialog box will
appear.
b) On the screen that appears, select "Stats" and press ENTER.
c) Enter data opposite positions as follows:
σ: 4.61, x¯ :15.58, n:90, and C-Level: .95.
d) Press ENTER, and the interval (14.628,
16.532) will appear along with values of other parameters.
Ex. 2: Suppose that you have a set of 35 temperature measurements
and you want to know with a 95%
confidence level what limits the population mean of temperature measurement will
fall within.
a) First you need to enter the data in a list, say list1, by
pressing 2ND, VAR-LINK, l (L, not 1), ENTER.
Enter you data into list1. Just enter a data point and press either ENTER or the down
arrow. Leave
the cursor in in the list that your data is in.
b) Press 2ND, F2, 1.
c) Select "Data" on the dialog box that appears and press ENTER.
d) Next you need to know the sample standard deviation. To enter
that opposite σ, do this: Press 2ND,
MATH, 6. You should now have stdDev( in the box opposite σ.
e) Press 2ND, VAR-LINK_{ }, l (L, not1), ENTER. You should
now have stdDev(list1 pasted in the box
oposite σ. Close the parentheses and move down to "List."
f) Enter information as follows:
List: If the correct list is not entered, press 2ND, VAR-LINK, l (L, not 1),
move to the list you want and press ENTER. Move down and enter Freq: 1, C-Level: .95.
Be sure
that the two list names are the same.
g) Press ENTER. The same type data will be
displayed as in Ex. 1 above.
VII. Other Distributions and Calculations:
1.
TInterval: If the sample size is <30, then the sample mean cannot be used for the
population mean, and
the ZInterval cannot be used. However, if the distribution is essentially normal, i.e.,
known to be normal
form other sources or has only one mode and is essentially symmetrical, then the Student t
Distribution
can be used.
Ex. 1: Suppose you take ten temperature measurements with sample mean x¯
= 98.44 and s = .3.
What is the 95% confidence level interval?
a) From the lists screen, press 2ND, F2 (for F7), 2.
b) On the screen that appears, use the cursor to to select "Stats" and press ENTER.
c) Enter data opposite positions as follows:
x¯ :98.44, S_{ x} : .3_{ }n:10, and C-Level: .95.
d) Press ENTER, and, after a few seconds, the interval {98.23,
98.65} will appear along with the values
for
"n" and the mean and a few other parameters..
Ex. 2: Suppose that you have a set of 10 temperature measurements
and you want to know with a 95%
confidence level what limits the population mean of temperature measurement will
fall within.
a) First you need to enter the data in a list, say list1_{,} by
pressing the "Stat/List icon. Now press
ENTER and entering your data in the list that appears. Just enter a data point and press either
ENTER or the down
arrow.
b) Press 2ND, F2, 2 to bring up the "Choose Input Method" screen.
c) Use the cursor to set to to "Data" on the screen and press ENTER.
d) On the TInterval screen that appears, enter information as follows:
List: Press 2ND, VAR-LINK, l (L, not 1).
If the cursor is not on list1, move it to list1 and press ENTER. This will
place list1 in the List box of the TIntrval
screen. Now enter 1 opposite Freq: 1 and .95 opposite C-Level.
e) Press ENTER. After a few seconds, the
interval (xx.xxx, xx.xx) will appear along with the values for
"n," the mean, and sample standard
deviation.
2. Student's t Distribution: The Student's t Distribution is
applied similar to the normal probability function, but it
can be applied where there are less than 30 data points, for example: P(t>
1.4|df = 19). The last part means
that the number of degrees of freedom ( one less that the number of data points)
is 19.
Ex. 1: Find the probability that t> 1.4 given that you have
20 data points.
a) Press F5, 6, 5, to bring up the tcdf( dialog box.
b) Enter data in the boxes as follows: Lower Value: 1.4, Upper Value
1E9, Deg of Freedom, df: 19.
c) Press ENTER and your answer should be .0888.
3. invT: Finding a t-value Given
α
and df:
This calculator has an invT, so do the
following:
(1) Press F5, 2, 2, and the Inverse t dialog box will appear.
(2) Enter α or 1-α, depending
on whether you have a left or right tail; then enter the degrees of freedom, df.
(3) Press ENTER and the value for "t" will be displayed.
Note that you may need to divide α by 2 if you
have not already made that adjustment.
4. The Chi-squared Distribution:
The χ^{2} Distribution is implemented
similar to the Student's t
distribution.
Ex. 1: Assume that you want to find P(χ^{2} >
24|df=20) the same as in the above Student's t Distribution.
a ) Press F5, 8 to bring up the Chi-Square Cdf dialog box.
b) Enter data so that your display is as follows:
Lower Value: 24, Upper Value: 1E9, df: 19.
c) Press ENTER and your answer should be .1961.
VIII. Hypothesis Testing:
1. Testing for Mean and z
Distribution with Data:
a) Enter
the data into list1_{ }or whatever list you choose.
b) Press 2ND,
F1, 1, ENTER
and move the cursor over to TESTS.
c) When the
"Input Method" dialog box appears, select "Data" and press ENTER.
e) Opposite
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,
MATH, 6, and stdDev(, will be displayed
opposite σ. Now, enter your
list number where the data is stored by pressing 2ND, VAR-LINK, l, (L, not1),
select the appropriate list
and press ENTER.
g) Opposite
"List," enter the list where you data is stored with the keystrokes mentioned
above and 1
opposite Freq.
h) Select the
proper condition for the alternative hypothesis.
i) Move
the cursor to Calculate and press ENTER.
j) If
you want to use the calculator to find the z-value or critical value, see those
procedures below.
2.
Testing for Mean and z Distribution with Statistics:
a) Select the Stats/Lists icon and press ENTER.
b) Press 2ND,
F1, 1 or
ENTER for Z-Test.
c) Select "Stats"
on the dialog box that appears and press ENTER.
d) Opposite
µ_{o}, enter the mean for the null hypothesis.
e)
Enter the given values for σ, x-bar, and n.
f) Select the
proper condition for the alternative hypothesis.
g) Move
the cursor to Calculate and press ENTER. The z-value, p-value and some
other statistics will
be displayed.
3. Finding a
z-vlaue for a particular confidence level:
Suppose you want
the z-value for a particular α,
e.g., 5%. Do this:
a) Press F5,
2, 1 for invNorm(.
b) Opposite
"Area," enter the value for α for a left-tailed or 1-α for a right-tailed. Clear the
values for
µ and σ or enter 0 opposite µ and 1
opposite σ.
c) Press ENTER and the z-value will be displayed.
4)
Finding critical values of x.
Suppose you have a
mean of 5.25, standard deviation of .6 and you want the critical number for an
α
of 5%.
a)
Press 2ND, F1, 1, and invNorm( will be pasted to the home screen.
b)
Enter the value for area, e.g., .05, µ: 5.25, σ: .6.
For a left tail, enter the value
for α and for a right tail enter 1-α..
c) Press
ENTER and the inverse will be displayed.
5. Testing for Mean
and t Distribution with Data:
a) Enter
the data into list1_{ }or whatever list you choose.
b) Press 2ND,
F1, 2 for T-Test.
c) On the
dialog box that appears, select "Data" and press ENTER.
e) Opposite
µ_{o}, enter the mean for the null hypothesis.
f) Enter
list1 or whatever list your data is in and opposite Freq. enter 1.
g) Select the
proper condition for the alternative hypothesis.
h) Move
the cursor to Calculate and press ENTER.
i) If
you are working a problem using the p-value test, read the p-value and compare
it with α or α-1 as appropriate.
j) If
you are working a problem using the t-value test, you will need to know the
critical values for the level of
significance, α, that you have chosen.
Refer to the procedures directly above to find these values with a
TI-89 Titanium.
6.
Testing for Mean and T Distribution with Statistics:
a) Press 2ND,
F1, 2 for T-Test.
b) On the
dialog box that appear, select "Stats" and press ENTER.
c) Opposite
µ_{o}, enter the mean for the null hypothesis.
e)
Enter the given values for σ, x-bar, and n. If you don't know x-bar you can
enter it by placing the cursor opposite
the symbol for mean; then press 2ND, MATH, 6, 4 and Mean( will be pasted to the
box opposite mean.
f) Press 2ND,
VAR-LINK, l (L, not1), select the appropriate list and press ENTER. Close
the parentheses.
g) Select the
proper condition for the alternative hypothesis.
h) Move
the cursor to Calculate and press ENTER.
i) If
you are working a problem using the p-value test, read the p-value and compare
it with α or α-1 as appropriate.
j) If you are
working a problem using the t-value test, you will need to know the critical
values for the level of
significance, α, that you have chosen. You can find the
z-value or the critical x-value using the procedures
above in this section.
X.
Statistics of two Populations:
1. Confidence Interval for Two
Dependent Populations (Data):
Enter the data
from population 1 into "list1" and the data from population 2 into List2.
Do this as follows:
a) Press APPS, select the
"Stats/Lists" icon and press ENTER.
b) Enter the data in the
displayed lists.
Now, store the paired differences in list list3 as follows:
c) From the Home screen, press 2nd, VAR-Link, l
(L not 1), select list1 and press ENTER.
d) Press the minus sign; then 2nd, VAR-Link, l (L not
1), select list2 and press ENTER.
e) Press STO, 2nd, VAR-Link, l (L not 1), select
list3 and press ENTER. You should now have
list1-list2 → list3 on the
Home screen.
f) Press ENTER and the pared differences will be
stored in list3.
Now, find the confidence level as follows:
g) Press 2ND, F2, 2 TInterval.
h) On the screen that appears, press the right cursor arrow
and select "Data" and press ENTER, ENTER.
i) In the dialog box that appears, enter
list3 in the box, either by using the VAR-LINK method or by typing
list3 by hand.
j) Enter 1 opposite "Freq," and the confidence
level you want, for example .95, opposite "C-Level."
j) Press ENTER and perhaps ENTER again and the confidence
interval and other statistics will be
displayed.
2. Confidence Interval for Two
Dependent Populations (Stats):
If you do not have data, but have the mean,
standard deviation, and n, use this procedure.
Find the confidence level from the list screen as
follows:
a) Press 2ND, F2, 2 TInterval.
b) On the that appears, press
the right cursor arrow and select "Stats" and press ENTER,
ENTER.
c) In the
dialog box that appears, enter the sample mean, the sample standard deviation,
the number of
data points, and the confidence level you desire.
d) Press ENTER and
perhaps ENTER again and the confidence
interval and other statistics will be
displayed.
3. Confidence Interval for Two
Independent Populations (Stats):
Find
the confidence level from the list screen as follows:
a) Press 2ND, F2, 4 for
"2-SmplTInt."
b) On the that appears, press
the right cursor arrow and select "Stats" and press ENTER and
ENTER
again if a new dialog box does not appear.
c) On the
dialog box that appears, enter the sample mean, the sample standard deviation,
the number of
data points, for both samples. In the last space, enter the confidence level you desire.
d) Press ENTER and
perhaps ENTER again and the confidence
interval and other statistics will be
displayed.
4. Confidence Interval for Two
Independent Populations (Data):
First, we must enter the data from population
lists.
Do this as follows:
a) Press APPS, select the "Stats/Lists" icon and
press ENTER, STAT, ENTER.
b) Enter the data in the
displayed lists, for example list1 and list2..
Now, find the confidence level as follows:
c) Press 2ND, F2, 4
for "2-SmplTInt."
d) On the screen that appears, press the right cursor arrow
and select "Data" and press ENTER, ENTER.
e) In the dialog box that appears, enter
list1 in the box after "List 1," list2 after "List 2;" enter 1 in both the
"Freq 1" and "Freq 2"
boxes. You can enter the list numbers either by using the VAR-LINK lists
or by
typing the lists in by
hand.
f) Enter the confidence level you want, for example
.95, opposite "C-Level."
h) Highlight
"No" opposite "Pooled" if there are no assumptions about the variations.
i) Press ENTER and if the screen does not change,
press ENTER again.
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