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Brief User Guide for Casio CFX-9850 Plus and CFX-9750 Statistics
UNDER CONSTRUCTION, USE WITH CARE.
Only items in the index in blue font
have been modified for the Casio CFX-9850! Applicable topic headings are
also
in blue font.
INDEX:
To facilitate lookup, the instructions
are divided into the following categories:
I. Data
Manipulation - Entering data, sorting data, clearing lists,
editing previous entries, 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. 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,
binomialpdf,
binomialcdf.
VIII. Hypothesis testing - mean and z-test
(data), mean and z-test (statistics), mean and t-test (data),
mean and t-test (statistics).
IX. Simple program for
calculating InverseT with a TI-83Plus.
X.
Statistics of two Populations - confidence interval for two dependent
population,
confidence interval for two
independent populations (Data and Stats),
RELEASE DATE: Not Released DATE LAST REVISED:
11/2/08
Printer friendly page
here.
NOTE: Copying
limitations and printing hints are at the end of 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, List 1 -List 6 is to place the
cursor on the name above the list and press F4(Del-A); then F1(YES).
1)
Entering Data:
a) Press MENU,
highlight the STAT icon; then
press EXE. 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 F3(DEL). Incidentally, you
can't delete an item in a list
with the DEL key.
2) Putting Data in Order:
a) To sort in
ascending order, highlight the list name and press F2(SRT-A). You will be
prompted
with "H, how many?." Enter 1 or however many you want to sort. You
will then be prompted with
Select list. Enter the number(s) of the list(s) you want to sort.
b) Note that you can also sort data in descending order with
using the same procedure with F2(SRT-D).
3) Editing Previous Entries:
Let's say you've entered
several calculations and you want to edit the last one. Press the left
cursor if want
to start editing at the
end of the expression or the right cursor if you want to start at the beginning.
Now, suppose you have
entered three expressions, and you want to go back and edit the first one.
a) Press
AC. All of the expressions are cleared from the screen, but don't
panic. They are all stored in
memory.
b) Press
the up arrow and the expressions and will appear in inverse sequence
as the up-arrow is
pressed
sequentially.
4) 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 2x4.7 = 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 List 1, go to that list. Make sure your data is
sorted in ascending order.
If you have frequencies in List 2 for example, be careful about sorting, or you
will get your
frequencies out of synch with the data values. Do the sorting of two lists as
follows:
(1) While on the List screen, press F1(SRT-A). In response to the
prompt "How Many Lists?
H," enter 2, EXE.
(2) When "Select Base List (B)" appears enter 1, EXE.
(3) In response to the prompt "Select Second List (L)," enter 2, EXE.
e)
Now, scroll down to the last number that falls within the upper limit of the first
class. At
the side of the list your will see
a number, where the is the number of data items in
your first
class.
f) 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 at the side. 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.
g)
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.
h)
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 CFX-9850 Plus:
This procedure
describes how to do a simple histogram for which the
calculator selects the class
width and,
therefore, the number of classes..
First you need to get your data into lists.
a) Go to the
STAT screen by pressing MENU, highlighting the STAT icon, and pressing EXE;
then enter your data.
b) Press
GPH1; then SET and highlight Graph Type and press F1(Sact). Enter List 1
opposite
XList by pressing F1. Enter List 2 opposite Frequency by pressing F3.
c) Press EXE
and then F1(GPH1).
d) On the
screen that appears, enter the number where you want the histogram to start
pitch. The pitch is the number of spaces that the bar for a class will
occupy.
e) Press Draw
and the histogram will be displayed. You can press SHIFT, Trace and use
the right/left
arrows to read the values of the classes.
3) Selecting Your Own Class Widths for the
Histogram Generated by the Calculator.
a) Enter your data into List List 1. If
your data is not in order, you can sort it by pressing STAT,
selecting SortA(,
then entering the list name of the data (often List 1).
As an example, you might
have this displayed on your screen: SortA(List 1.
Now, press ENTER, and your data will be sorted.
b) Now, from the sorted data, determine the class width and lower boundary
of the lowest class as
described under "Doing a Frequency Distribution by Hand" above.
Now do this:
c) Press [2nd], [STAT PLOT], highlight 1, and press [ENTER].
d) Highlight ON on
the next screen; then highlight the histogram symbol. Make sure List
1 is entered
for the Xlist.
Note that if you do not have tables data ( where frequencies are given),
use the default
value, 1, for the frequency.
e) Press
[WINDOW], and enter the lower boundary of the lowest class
as Xmin and your
chosen class width in Xscl.
Note that once the lower boundary and class width are set, the
upper limit is automatically
determined. Set Ymin at zero.
f) Press [GRAPH] and the histogram will appear. You can use [TRACE] to display the value
of the boundary limits and frequencies of a
particular bar on the histogram.
g) If the
graph extends above the top edge of the screen, Press WINDOW and increase the
Ymax
value. I also usually set Ymin to -1.
4. Constructing a Frequency Polygon from Ungrouped Data:
After graphing the histogram, you can use TRACE to get the data
for the frequency polygon and a cumulative
frequency graph if you wish.
a) Press TRACE and use the arrow to move across the histogram bars.
Record the values for 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
List 2 if you have your histogram data in List 1. 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 on the first row; then enter List 2
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 List 2. To do that place the cursor
at the first item in List 2, press INS and replace the zero that
appears with your the first midpoint you calculated. Go to the bottom of the List
2 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 List 2.
C. Proceed with step c) above.
5. Constructing a Cumulative Frequency Chart (Ogive) Graph:
a) Enter the Xmax values that you recorded above in a list. For
example, L4 if you still have data in the
other lists.
b) Now, store the cumulative frequency data in L 5 as
follows: Press 2nd, LIST, cursor to OPS, and press 6. cumSum(
will be posted to the home screen.
c) With the cursor after the parenthesis, press 2nd, L3,
), STO, 2nd, L5, ENTER. You will now have cumSum(L3)→L5
pasted to the screen.
d) Press 2nd, STAT PLOT, highlight "On" if necessary and press ENTER
e) Highlight the second icon on the first row; then enter L4
opposite Xlist and L5 opposite Ylist.
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.
f) Press ZOOM, 9 and the graph will appear on the screen.
6) Relative Frequency polygon and Cumulative Relative Frequency (Ogive)
Graphs:
Do
these exactly as in the frequency
polygon and cumulative frequency graph above except that after storing
the data (step b for the frequency polygon) do this
step: Press 2nd, L3 /N, STO, 2nd, L3 . This will convert
the data in L3 to relative frequency.
7) Histogram Using Grouped Data:
a) Enter
the midpoints of the classes into List 1 and the corresponding
frequencies into List 2 .
b) Press 2nd, STAT PLOT, ENTER.
c) If "On" is not highlighted, select it and press ENTER.
d) Move the
cursor to the histogram symbol and press ENTER; then enter List 1 opposite Xlist
and List 2 opposite Ylist.
e) Press ZOOM, 9 and the histogram will be displayed.
Note: If you want to select your own classes
do this before pressing ZOOM 9 in step "e" above.
1) Press WINDOW and enter
the lowest boundary value opposite Xmin
and the class width opposite Xscl.
You may also want to change Ymin to something like zero or -1 so that
histogram will not
be so far above the baseline.
2) Press GRAPH and the histogram will be displayed.
8) Frequency Polygon Using Grouped Data:
Do this exactly like the histogram, except select the line graph
icon, the second icon. If you've already done the
histogram, just change the icon and press GRAPH.
9) Cumulative Frequency (Ogive) Graph from Grouped Date:
a) Enter the upper class limits in a list, for example, L3
if you have data in the first two lists.
b) If you
have the frequency in List 2 , do the following:
A) Press 2nd, LIST, cursor to OPS, and press 6. cumSum( will
be posted to the home screen.
B) With the cursor after the parenthesis, press 2nd, List 2,
), STO, 2nd, L4 . You will now have
cumSum(List 2) →L4
pasted to the screen. Press ENTER.
c) Press 2nd, STAT PLOT, ENTER. If "On" is not highlighted, select
it and press ENTER.
d) Highlight the second icon, and enter 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 List 1 and the corresponding frequencies in
List 2.
If you want the graph to start
at zero, enter the first lower boundary with zero for the frequency.
b) Press 2nd,
QUIT to get out of the List.
c) Press
2nd, LIST, cursor to OPS, and press 6 to paste cumSum( to the home screen.
d) Press 2nd,
List 2 , ), ÷ . You now should have cumSum(List 2)/ on the
home screen.
e) Press 2nd,
LIST, cursor to MATH and press 5 to paste sum( to the screen.
f) Press
2nd, List 2, ). You now should have cumSum(List 2)/Sum(List
2)
on the home screen.
g) Press x
(the multiply symbol), 100, STO, 2nd, L3. You now should
have
cumSum(List 2)/Sum(List 2) x100→L3
pasted to the home screen.
h) Press
ENTER and the data will be stored in L3 .
Plotting the Data:
i) Press 2nd,
STAT PLOT, ENTER
j) Select the
second icon and enter List 1 opposite Xlist and L3 opposite
Ylist.
k) Press
ZOOM, 9 and your graph will be displayed.
l) You can
find the exact percentiles of the boundaries by using TRACE, and approximate
percentiles of
other 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 List 1. (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 5, and
press [ENTER] to highlight the box plot icon.
(See the note at the end of this topic for when to
use icon 4.)
f) Enter the list you put
the data in, usually List 1, in the Xlist, by pressing 2nd, List
1.
or whatever list
you chose.
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.
NOTE: If you have one or
two outliers (numbers much larger than the rest) you may want to use
icon 4.
This will not include the outliers in the last whisker, but will plot them as
separate points
after
the end of the last whisker.
13) Box and Whisker Plot by Hand
You can save yourself considerable
calculation if you use the calculator to find 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 List
1
just press ENTER. Otherwise press 2nd and the list name where your
data is stored.
c) Cursor down and you will find
Q 1 , 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 List 1 and the corresponding P(x)
values into List 2 as
the frequency. The details are as follows:
a) Enter the number
of flashes in list List 1 and the corresponding P(x) values in List
2
opposite the
number of flashes. (How to enter data in a list is covered at the
beginning of this document.)
b) Press STAT, cursor to
CALC and press ENTER. 1-Var Stats will be displayed on the home
screen.
c) Press 2nd, List 1,
press the comma, then 2nd, List 2 . You should now have
1-Var Stats List 1, List 2 on the
home screen.
d) Press ENTER and the
values for the mean (expected value), standard deviation and other
statistics will be displayed.
e) If you need the
variance, merely re-enter the value for the standard deviation, σx
, and square it,
15) Doing a Discrete Probability Distribution by Hand
Many teachers still see
value in cranking out the numbers for these statistics, so
here are methods
to take some
of the drudgery out of doing the arithmetic.
The mean can be obtained by the following formula: mean =
Σxp(x).
To obtain the
individual values and store them in list L3, do the following:
(The x-values should
should
be stored in List 1 and the p(x) values in List 2.)
a)
Press 2ND, List 1, x, 2ND, List 2, STO, 2nd, L3.
You will now have List 1xList 2→L3 pasted to the home
screen.
b)
Press ENTER and you will have the individual values stored in list L3
and displayed on the
home screen.
c)
To get the sum of these values, do this.
(1)
Press 2nd, LIST; cursor to MATH, and press 5. The expression sum( will be
pasted to
the home screen.
(2) Press 2ND, List 1 ,x, 2ND, List 2 , ), STO, 2ND,
L3. You will have sum(List
1
xList 1)→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) Press 2ND, List 1, x2, ,x, 2ND, List 2 , STO,
2ND, L3. You will have List 12xList
2→L3
pasted to the home
screen.
b) Press ENTER and the individual values will be entered in list L3
and pasted to the home
screen.
c) To get the sum of these values do the following:
(1)
Press 2nd, LIST; cursor to MATH, and press 5. The expression sum( will be
pasted to
the home screen.
(2) Press 2ND, Lust 1 ,x2 ,x, 2ND, List 2
, ), STO, 2ND, L3. You will have sum(
(List 1)2xList 2)→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(List 12xList
2)
- (sum(List 1
xList 2))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 List 1, and press MENU, highlight the RUN icon and press EXE.
QUIT to leave the lists.
b) Press 2nd, LIST and
move the cursor to MATH.
c) Press 3 to paste mean(
to the home screen.
d) Press 2nd, List 1, ),
and then press the divide symbol.
e) Press 2nd, LIST, move
the cursor to MATH, and press 7.
f) Press 2nd, List 1
and then press ENTER to display the CV.
17.
Finding the Standard Deviation and Variance of Ungrouped Data:
A. Calculated by
the Calculator Only
a) Entering Data:
1) If not at the
STAT screen, press MENU; highlight the STAT icon and press EXE.
Tables for entering data will appear. If
you need to clear a list, move the cursor up
to highlight the list name; then press
F6, F4(DEL-A), F1(YES)..
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
F3(DEL).
b) Suppose that you have the sample of data listed immediately below and
you want to find
the standard deviation and variance.
Data: 22, 27, 15, 35, 30, 52, 35
c) Enter the data in list List 1 as described under Entering Data
immediately above, then press
F2(CALC).
(If CALC is not at the bottom of the screen, press F6 first.
d) Press F1(1VAR) and the various
statistics will be displayed.
B.
Calculating Numbers to Plug into a Computation Formula:
The standard deviation can be found easily by using 1VAR
Stats as described above, but
many teachers require that students do the calculations by hand to learn
the details of the
process. The following give a method for
using the Casio CFX-9850 Plus 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) (List 1) |
x2 (List 2) |
|
1 |
22 |
484 |
|
2 |
27 |
729 |
|
3 |
15 |
225 |
|
4 |
35 |
1225 |
|
5 |
30 |
900 |
|
6 |
52 |
2704 |
|
7 |
35 |
1225 |
|
|
|
|
|
n=7 |
Σx=216 |
Σx²=7492 |
The variance
computation formula is as follows: 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 in item B immediately above.
b) Press F2(CALC),F6(SET). On the screen that appears
press F1(List 1) for the 1Var XList, move the
cursor to 1Var Freq and press 1. Press
EXE and F1(1Var). .
c) From the screen that appers, copy n=7, ∑x = 216, and ∑x2
=7492 and xσn-1=11.73.
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
and we will need to go to the RUN screen to calculate that. Press
MENU,
highlight the RUN icon
and press EXE.
e) Press OPTN, (, press F1(List),
x2, →, F1(List), 2, Now press EXE and "Done" will appear.
You can now
find the numbers for x2
in List 2.
f) Now, we want to use the numbers that we
previously recorded to plug into the variance
formula. So, at the RUN screen enter
(7492-2162÷7)÷(6).
g) Press ENTER and you should have 137.8…, which is the variance.
h) Recall that you copied the standard deviation
above, but if you want to calculate it, press SHIFT,
√ , SHIFT, Ans, EXE, and
you will have 11.73...
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 (List 1) |
Freq. (f) (List 2) |
|
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 List 1. You
can either do the midpoints by hand or calculate
and store
them in list List 1 as follows:
(1) Store the lower
boundaries in list List 1 and the upper boundaries in List 2.
(2) Press 2ND,
QUIT to get out of the list editor and press (, 2ND, List 1, + 2ND, List
2,), divide
symbol, 2 STO, List 1. You should have (List 1 + List
2)/2→
List 1 on the home screen. Press
ENTER and the midpoints will be stored in List 1.
d) Enter the frequencies in List 2 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,
List 1
; then press the comma and finally
press 2nd,
List 2.
e) Press ENTER, and the standard deviation along with several other
statistics will be
displayed. The sample standard deviation is 14.868….
f) To find the variance, just square the standard deviation by
entering the number, pressing
the 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 (List 1) |
Freq. (f) (List 2) |
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 List
1
as follows:
(1) Store the lower boundaries in list List 1
and the upper boundaries in List 2.
(2) Press 2ND, QUIT to get out of the list editor and
press (, 2ND, List 1, + 2ND, List 2,), divide
symbol, 2 STO, List
1. You should have (List 1 + List 2)/2→ List 1 on the home screen. Press
ENTER
and the midpoints will be
stored in List 1.
b) Press STAT, ENTER to go to the lists and store the frequencies in list
List 2.
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, List
1 ;
then press the comma and finally
press 2nd, List 2.
d) Press ENTER and several statistics along with the standard deviation will be
displayed.
Record the standard deviation, Sx =14.868 for a reference. Also record
∑x=∑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 by pressing 2ND, List 1, x, 2ND, List 2, STO, L3.
You should have
List 1xList 2→L3 on the home screen. Press
ENTER and the products will be stored in list L3 and will
be displayed on the home screen.
f) Calculate x2f by pressing 2ND, List 1, x2
, x , 2ND, List 2, STO, 2ND, L4 . You should now have
List 12 xList 2→L4 on the
home screen.
g) Press ENTER and the results will be stored in list L4 and will
be displayed on the home
screen.
h) You
don’t need to calculate
Σf. That is the value for “n” that you previously recorded.
i)
You don’t need to
calculate
Σxf. That is the value for ∑ x that you previously recorded.
j) Now,
you want to plug the appropriate numbers into the formula for the variance. From
the
home screen enter
(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 and Regression analysis
finding a, b, r, and r2.
First you need to get your data into lists.
a) Press MENU, highlight the
STAT icon and press EXE. If you want to clear the lists first, lighlight the
list
name, for example List 1, and press F3(Del-A); then pressF1 (YES).
Now, enter the data by
positioning the cursor in the list, entering a data point, and pressing EXE.
b) Press
F6(SET) and on the screen that appears, move the cursor to Graph Type and press
F1 (Scat). Move
the
cursor to XList and press F1(List 1). In the same manner, enter List2 or
YList, 1 for Frequency, and
choose
the Mark Type and Color as you wish. Press EXE to return to the list
screen.
c)
Press F1(GRPH1) and the scatter points will be entered.
To get a, b, r and r2
proceed as follows:
a) Press F1(x) and those parameters will be
displayed. Note carefully that the equation is y=ax+b. So, a is
the slope and
b is the y-intercept with this calculator. This is different from many
statistics books which
have y=a+bx
where y-intercept and b is the slope.
e) To plot the graph, from that same
screen, press F6(Draw).
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
F2(xyLine) rather than F1(Scat).
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.
First you need to get your data into lists.
a) Press MENU, highlight the
STAT icon and press EXE. If you want to clear the lists first, lighlight the
list
name, for example List 1, and press F3(Del-A); then pressF1 (YES).
Now, enter the data by
positioning the cursor in the list, entering a data point, and pressing EXE.
c) Press F6(SET), choose the proper lists,
say List 1 and List 2; then
press EXE to return to the list
screen.
d) Press F2(2VAR). and the
appropriate values will be displayed. 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. Although it is not
always necessary, I recommend that all square
roots of more than one term be enclosed in parentheses.
Example: √(nΣx2- (Σx)2).
Now let’s plug the numbers into the equation for r:
e) Press MENU, highlight
the RUN icon and press EXE.
f) r= (nΣxy
–ΣxΣy)/[(√(nΣx2- (Σx)2)(√(nΣy2- (Σy)2)]
= (8x62750-521x960)/(√(8x33979-5212)(√(8x116900-9602))
=.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:
8x62750-521x960, →, ALPHA, N.
2) Calculate the
denominator and store it in two separate variables M and D. In this manner
√(8x33979-5212 ) , →, ALPHA, M; then √(8x116900-9602),
→, ALPHA, D.
3) N÷(MxD), 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 |
First you need to get your data into lists.
a) Press MENU, highlight the
STAT icon and press EXE. If you want to clear the lists first, lighlight the
list
name, for example List 1, and press F3(Del-A); then pressF1 (YES).
Now, enter the data by
positioning the cursor in the list, entering a data point, and pressing EXE.
b) Press
F6(SET) and on the screen that appears, move the cursor to Graph Type and press
F1 (Scat). Move
the
cursor to XList and press F1(List 1). In the same manner, enter List2 or
YList, 1 for Frequency, and
choose
the Mark Type and Color as you wish. Press EXE to return to the list
screen.
c)
Press F1(GRPH1) and the scatter points will be entered.
To get a, b, r and r2
proceed as follows:
a) Press F1(x) and those parameters will be
displayed. Note carefully that the equation is y=ax+b. So, a is
the slope and
b is the y-intercept with this calculator. This is different from many
statistics books which
have y=a+bx
where y-intercept and b is the slope.
e) To plot the graph, from that same
screen, press F6(Draw).
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 List 1
and List 2 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-83
Plus or TI-84, 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).
=(8x62750-521x960)/(8x33979-5212)
=4.7058…..
e) Now, calculate the value for a from the formula:
a= y-bar –b(x-bar)
=120-4.7058 x65.125
=-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:
8x62750-521x960, STO, ALPHA, N.
2) Enter the
denominator and store it in variable D. 8x33979-5212 , 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, List 1 (or whatever list you want to sort); then press ENTER.
"Done" will be displayed
on the home screen,
indicating your data has been sorted. Note that you can also sort data in
descending order with SortD.
2. Finding Mean
(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 List 1.
c) Press ENTER.
d) A number of results will be displayed on the home
screen.
NOTE: You can also find these values for
discrete random variable statistics by entering the values
of the variable in List 1 , for example, and the corresponding data
values in List 2.
For Grouped data:
a) Find the midpoints of each group and enter those
values in List 1; then enter the corresponding frequencies
List 2. Entering data in
a list is described in
item 1 of Section I, above.
b) Press STAT, cursor over to CALC, and press ENTER.
"1-Var Stats" will be pasted to the home screen.
c) Press 2nd, List 1, 2nd, List
2;
then press ENTER.
d) Various statistics will be displayed on the home
screen. Note that for grouped data, ∑xf is listed on the
calculator as ∑x and ∑x2
f is listed as ∑x2 .
3. Finding products
such as xy or (x-y):
a) Assume that your x-data is in List 1 and
your y-data is in List 2. Then obtain the product by pressing
2nd, List 1; x (multiply
symbol), 2nd, List 2, ENTER.
b) If you want the data stored in a list, L3
for example, before pressing ENTER in item a, press 2nd,
List 1, 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 List 1.
b) Press 2nd, List 1; 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 before pressing ENTER in
item b above, press 2nd, STO,
2nd, L3. Then press ENTER.
d) If you want to multiply
corresponding elements of two lists and square each result; then your
expression should be like this:
(List 1 x List 2)2 .
5. Find x-x¯ (Sorry,
I have no symbol for the mean, so I displaced the bar.) from the data in
list List 1.
a) Enter 2nd, List 1, -, 2nd, LIST.
Note that" -" is a minus sign not a negative sign.
b) Cursor to MATH and press 3. You should
now have "List 1-mean(" pasted to the home screen.
c) Press 2nd, List 1, 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, List 1, -, 2nd, LIST.
b) Cursor to MATH and press 3. You
should now have "(List 1-mean(" pasted to the home screen.
c) Press 2nd, List
1,),),x2 . The expression ((List 1-mean(List
1))2 should now be
displayed on the screen.
Press ENTER and the
results will be displayed on the home screen.
d) If you want to store the results in a
list, for example 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, List 1 or whatever
list your data is stored in.
e) Press ), ), x2 . You
now should have (sum(List 1))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
List 12 "
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.
Suppose we want the
permutations (arrangements) of 8 things 3 at a time.
a) If not at the RUN screen, press MENU, highlight the
RUN icon and press EXE. Enter 8 on the home
screen.
b) Press OPTN, F6, F3(PROB), F2(nPr), 3.
c) Press
EXE. You will get 336.
2.
Finding Combinations:.
Suppose we want the
combinations (groups) of 8 things 3 at a time, enter 8 on the home screen.
a) If not at the RUN screen, press MENU, highlight the RUN icon and
press EXE. Enter 8 on the home
screen.
b) Press OPTN, F6, F3(PROB), F3(nCr), 3.
c) Press
EXE. You will get 56.
3. Finding Factorials.
Suppose we want 5 factorial (5!). From
the RUN screen press 5.
a) If not at the RUN screen,
press MENU, highlight the RUN icon and press EXE. Enter 5 on the home
screen.
b) Press OPTN, F6, F3(PROB), F3(x!).
c) Press
EXE. You will get 120.
4. Randomly
generated data sets:
Sometimes problems use a randomly generated set
of data. Suppose we want to generate 10
random numbers between 1 and 50 and store them in
List 1. The proper syntax is randint(lower,
upper, how many). That can be obtained
as follows:
a) Press MATH, cursor over to PRB and press
the number 5. randint( will appear on the screen.
b) Enter 1, 50, 10, so that your screen
displays randint(1,50,10). Press ENTER
c) Now, if you want to cause these numbers
to be stored in List 1, before pressing ENTER in item b,
press STO;2nd, List
1. The entries, randint(1,50,10)->List 1,
will appear on the screen.
d) Press ENTER and the numbers generated
will appear on the screen and will be stored in list List 1.
VI.
Normal Distribution:
Note:
In this section, a general method will be
outlined; then a specific example will be worked. The
same
problem will be used in several of the examples.
General, normalcdf(: This function returns the value of the area between two
values of the random variable
"x." This can be interpreted as the probability that a randomly selected variable will fall
within those two
values of "x," or as a percentage of the 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) Press 2nd, DISTR, 2.
The term "normalcdf(" will appear on the home screen.
b) Enter the number for the
left boundary, right boundary, μ, and σ in that order. You do not need
to
close the parentheses, but it's okay if you do.
c) Press ENTER and the value of
the area between the two points will be displayed. Notice that
you do
not explicitly convert the points to 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 so that your display is the following: normalcdf(65,80,70,4.5.
c) Press ENTER and
you'll get 0.85361 which is, of course, 85.361 percent.
2.
normalcdf(: Area under a curve to the left of a point with μ (mean) and σ (std.
dev.) given.
Ex. 2: In the
above problem, determine the probability that the value is less than 62.
a) Complete
item a) in the general method above.
b)
Enter numbers so that your display is the following: normalcdf(-1E9,
62,70,4.5. Notice that
the "-" is a negative sign, not a minus sign. Enter "E" by pressing 2nd,
EE (The comma
key.)
c) Press
ENTER and you'll get 0.03772 which is, of course, 3.772 per percent.
3.
normalcdf(: Area under a curve to the right of a point with μ (mean) and σ (std. dev.)
given.
Ex. 3: In the
above problem, determine the probability that a value is greater than or equal to
75.
a) Complete
item a) in the general method above.
b)
Enter numbers so that your display is the following: normalcdf(75,
1E9,70,4.5.
Enter "E" by pressing 2nd, EE (The comma key.)
c) Press
ENTER and you'll get 0.13326 which is, of course, 13.326 per percent.
4. ShadeNorm(: Displaying a graph of
the area under the normal curve.
General:
This function draws the normal density function specified by µ and
σ and shades the area
between the upper and lower bounds.
This is essentially a graph of normalcdf(. It will display the
area and
upper and lower bounds. Not including µ and σ defaults to a normal curve. The following
instructions, "a" through "c," are general instruction to follow.
a) First
turn off any Y= functions that may be active. Do this by moving the cursor
to a
highlighted = sign and pressing ENTER.
b) Press 2nd,
DISTR and cursor over to DRAW. Press 1 and ShadeNorm( will appear on the
home screen. Enter the correct parameters depending on whether the problem
is like 1, 2,
or 3 above.
c) Press
ENTER, and the graph may be visible on the screen. You will almost
certainly need
to reset the Window parameters by pressing WINDOW and changing Xmin, Xmax, Ymin,
and
Ymax settings to get a decent display. As a first approximation, set Xmin at 5
standard
deviations below the mean and Xmax at 5 above the mean. (See the following
example.) Start out with
a Ymax about 0.3 and go from there. You can set the Ymin at 0, or if you wish, set it at
about
negative one-fiftieth of Ymax. You may need to fine tune from there.
Ex 1:
Draw the graph of example 2 above.
a) Press WINDOW and set Xmin=50, Xmax=90, ymin=-.01, Ymax = 0.1. You
can reset the
scales as you choose to eliminate the broad baseline.
b) Press 2nd, DISTR and cursor over to DRAW. Press 1 and ShadeNorm(
will appear on the
home screen.
c) Enter parameters so that your display looks like this: ShadeNorm(-1E9,
62, 70, 4.5.
d) Press ENTER and a reasonable looking graph should appear on the screen.
5. invNorm(: Inverse Probability Calculation:
Find the number x, in a normal distribution such that a number is less than x
with a given
probability.
The syntax for this is invNorm(area, [μ, σ]). The part in brackets
indicates that there
is a default for those values. The default is
the standard curve with mean=0
and standard deviation. is 1.
Ex. 1:
In Ex. 1 immediately above, find the number x, such that a randomly selected number will be
below
that
number with a 90% probability.
a)
Press 2nd, DISTR, 3 to select invNorm(.
b)
Enter parameters so that your display looks like this: invNormal(.90,70,4.5.
c)
Press ENTER and your answer will be 75.766.
Ex.
2: Given a normal distribution with a mean of 100 and standard
deviation of 20. Find a value Xo such
that the given x-value is below Xo is .6523. That is P(X<Xo)
= .6523.
a) Press 2nd, DISTR, 3 to place "invNORM(" on the home screen.
b) Enter information so that the entry looks like the following:
invNORM(.6523,100, 20.
c) Press ENTER and your answer will be 107.83.
Ex. 3: What is the lowest score possible to be in the upper 10% of
the class if the mean is 70 and the
standard deviation is 12?
a) Press 2nd, DISTR, 3. to place "invNORM(" on the home screen.
b) Enter information so that the entry looks like the following:
invNORM(1-.1,70, 12. Your answer will
be 85.38 or 86 rounded off.
6. ShadeNorm(: Window Settings for Graphing (shading) the Inverse Probability area:
General:
If you are accustomed to graphing using the standard WINDOW settings called by
ZOOM, 6, then you're in for a big surprise if you use those settings for
graphing the normal
curve. So, before you display the ShadeNorm( function, press WINDOW and
set the values
as follows:
a) Xmin =
μ - 4σ. Round of to the next
integer.
b) Add the same number to the mean that you subtracted from the Xmin to
get Xmax.
c) Xscl= Set at the standard deviation.
d) Ymin=0. Some people like to set this at a small negative number,
but if you have
problems with a wide range of std. devs. you'll have to keep changing it.
I set it at 0; then
I'm done with it.
e) Ymax= As a first approximation, set this at 0.4/σ.
f) Yscl= Most of the time the y-axis is not displayed, so I usually just
set it at 0.01 and
leave it there.
7.
ShadeNorm(: Graphing (shading) the Probability area:
Ex. 1: Obviously
if you wanted to graph the example immediately above, you could use the
ShadeNorm(
using the lower bound of -1E9 and the upper bound of 75.766. You would do
that
as follows:
a) Press WINDOW and
set Xmin=50, Xmax=90, ymin=-.005, Ymax = 0.1. You
can reset the
scales as you choose to eliminate the broad baseline.
b) Press 2nd, DISTR and cursor over to DRAW. Press 1 and ShadeNorm(
will appear on the
home screen.
c) Enter parameters so that your display looks like this: ShadeNorm(-1E9,
75.766, 70, 4.5.
d) Press ENTER and a reasonable looking graph should appear on the screen.
Note that if you wanted to shade the region where the probability would be
greater than 90%,
you would choose 75.766 for the lower boundary and 1E9 as the
upper bound.
Ex. 2:
Suppose you wanted to graph a distribution and shade the area between the points 40 and 54,
with a mean of 46
and a std. dev.
of 8.5
a) Press WINDOW and
set Xmin=12, Xmax=80, Ymin=-.005, Ymax = 0.06. You
can reset the
scales as you choose to eliminate the broad baseline.
b) Press 2nd, DISTR and cursor over to DRAW. Press 1 and ShadeNorm(
will appear on the
home screen.
c) Enter parameters so that your display looks like this: ShadeNorm(40,
54, 46, 8.2.
d) Press ENTER and a reasonable looking graph should appear on the screen.
The area
under the curve, 0.603198, will be displayed on the screen along with
the upper and lower
bounds.
8. normalpdf(: Probability Distribution Function using normalpdf( :
General: This function is used to find the fraction, and therefore
also the percentage, of the
distribution that corresponds to a particular value of x. The syntax of
this function is
normalpdf(X, μ, σ
A)
Finding the Percentage of a Single Value:
Ex. 1: Suppose that the mean of a certain distribution is 60 and the
standard deviation is 12.
What percentage of the population will have the value 50?
a) Press 2nd, DISTR, 1 to paste normalpdf( to the home screen.
b) Enter data so that your display is as follows:
normalpdf(50,60,12.
c) Press ENTER and your answer should be .02317 which is about 2.3
percent.
B)
Graphing the distribution:
Ex. 1: Suppose that
the mean of a certain distribution is 60 and the standard deviation is 12.
Investigate percentages for several 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.
b) Press Y= and select the Y1= position; then press 2nd, DISTR, 1 to paste
normalpdf( to
the Y1= position.
c) Enter data so that the entry after Y1= looks line this:
normalpdf(X, 60,12.
d) Press ZOOM, 0 to select ZoomFit and the curve should appear on the
screen.
e) Press TRACE and you can move along the curve and read the values for
different x-
values. If you want a specific value, perhaps to get rid of the x-value
decimals, just enter
that number and press ENTER.
9. ZInterval: This gives the range within which the population mean is expected to fall
with a desired
confidence level. The sample size should be > 30 if the
population standard devation is not
known.
Ex. 1: Suppose we have a sample of 90 with sample mean x¯ =
15.58 and s = 4.61. What is the 95%
confidence level interval?
a) Press STAT, cursor to TESTS, and press 7.
b) On the screen that appears, cursor to "Stats" on the ZInterval screen and press ENTER.
c) Enter data opposite positions as follows:
σ: 4.61, x¯ :15.58, n:90, and C-Level: .95.
d) Cursor down to Calculate, press ENTER, and the interval (14.628,
16.532) will appear along with
the values for "n" and the mean.
Ex. 2: Suppose that you have a set of 35 temperature measurements
and you want to know with a 95%
confidence level what limits the population mean of temperature measurement will
fall within.
a) First you need to enter the data in a list, say List 1, by
pressing STAT, ENTER, and entering your data
in the list that appears. Just enter a data point and press either ENTER or the down
arrow.
b) Press STAT, cursor to TEST and press 7 to get the ZInterval screen.
c) Cursor to "Data" and press ENTER.
d) Next you need to know the sample standard deviation. To enter
that opposite σ, do this: Press 2nd, LIST,
move the cursor to MATH and press 7. The expression stdDev( will be pasted
opposite σ.
e) Press 2nd, List 1 , or whatever list you have your data in. When
you move the cursor the value will be entered.
f) Enter information as follows: List: Press 2nd, List 1, Freq: 1, C-Level: .95.
g) Cursor to Calculate and press ENTER. The same type data will be
displayed as in Ex. 1 above.
VII. Other Distributions and Calculations:
1.
TInterval: If the sample size is <30, then the sample mean cannot be used for the
population mean, and
the ZInterval cannot be used. However, if the distribution is essentially normal, i.e.,
know to be normal
form other sources or has only one mode and is essentially symmetrical, then the Student t
Distribution
can be used.
Ex. 1: Suppose you take ten temperature measurements with sample mean x¯
= 98.44 and s = .3.
What is the 95% confidence level interval?
a) Press STAT, cursor to TESTS, and press 8.
b) On the screen that appears, cursor to "Stats" and press ENTER.
c) Enter data opposite positions as follows:
x¯ :98.44, S x : .3 n:10, and C-Level: .95.
d) Cursor down to "Calculate", press ENTER, and, after a few seconds, the interval (98.228,
98.655)
will appear along with the values for
"n" and the mean.
Ex. 2: Suppose that you have a set of 10 temperature measurements
and you want to know with a 95%
confidence level what limits the population mean of temperature measurement will
fall within.
a) First you need to enter the data in a list, say List 1, by
pressing STAT, ENTER, and entering your data
in the list that appears. Just enter a data point and press either ENTER or the down
arrow.
b) Press STAT, cursor to "TEST" and press 8 to get the TInterval screen.
c) Cursor to "Data" on the TInterval screen and press ENTER.
d) Enter information as follows: List: Press 2nd, List 1, Freq: 1, C-Level: .95.
e) Cursor to "Calculate" and press ENTER. After a few seconds, the
interval (xx.xxx, xx.xx)
will appear along with the values for "n," the mean, and sample standard
deviation.
2. Student's t Distribution: The Student's t Distribution is
applied similar to the normal probability function, but it
can be applied to where there are less than 30 data points, for example: P(t>
1.4|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 give that you have
20 data points.
a) Press 2nd, DISTR, 5, to paste tcdf( to the home screen.
b) Enter data so that your display is as follows:
tcdf(1.4, 1E9,19.
c) Press ENTER and your answer should be .0888...
3. invT: Finding a t-value Given
α and df:
If you are working a problem using the t-value, there are different options
depending on your needs and whether
you're using a TI-83 Plus or a TI-84 Silver Edition.
TI-84 Plus Silver Edition: This calculator has an invT, so do the
following:
(1) Press 2nd, DISTR, 4, and invT( will be pasted to the screen.
(2) Enter α or 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.
TI-83 Plus: This calculator does not have an invT, so you can do
either of two procedures:
(1) Look up the t-value in your book. This is by far the easier.
(2) If you have an α that's not in the table or don't have a table,
you can do this:
Suppose you want the t-value for α=.1 for a left-tailed test.
(a) Press MATH, 0, and the solver will be pasted to the screen.
(b) Press the UP arrow so that the equation is displayed.
(c) Press 2nd, DISTR, 5 and tcdf( will be pasted in as a formula.
(d) Enter data so that your entry will look like this: tcdf(-1E9, X, 10) -
.100 and press ENTER.
(e) Press the UP arrow and enter -1 opposite X.
(f) Press ALPHA, SOLVE, and the value for "t" will be displayed
opposite X after about 20 seconds.
Suppose you want the t-value for α=.1 for a right-tailed test.
The steps are exactly the same except for these.
(d) Enter data so that your entry will look like this: tcdf(-1E9, X, 10) -
.900 and press ENTER.
(e) Press the UP arrow and enter 1 opposite X.
Use a Calculator Program:
There are several program posted on the Web, for example, at
www.ticalc.org . I will also be
posting a
program that I have written some time soon. It may not be the greatest,
but it works.
4. The Chi-squared Distribution:
The χ2 Distribution is implemented
similar to the Student's t
Distribution.
Ex. 1: Assume that you want to find P(χ2 >
24|df=20) the same as in the above Student's t Distribution.
a ) Press 2nd, DISTR, 7, to paste χ2cdf( to the home screen.
b) Enter data so that your display is as follows:
χ2cdf(24, 1E9,19.
c) Press ENTER and your answer should be .1961...
7. Binomial Distribution,
Bpd:
Suppose that you know that 5% of the bolts coming out of a
factory are defective. You take a sample of 12.
Determine the probability that 4 of them are defective.
a) If you are not at the STAT screen, press MENU, highlight the STAT icon
and press EXE.
b) Press F5(DIST), F5(BINM), and F1(Bpd).
c) On the screen that appears, press F2(Var) to enter Variable opposite
Data.
d) Enter 4 opposite x, 12 opposite Numtrial, and .05 opposite p.
e) Press F1(CALC) to get 2.05E-03 for the answer. If you want to repeat, just
press EXE.
8. Binomial Distribution,
Bcd:
Suppose that you know that 5% of the bolts coming out
of a factory are defective. You take a sample of 12.
Determine the probability that 4 or more of them are defective. I'll show
you two different methods for
doing this problem.
First Method:
a) If you are not at the STAT screen, press MENU, highlight the STAT icon and
press EXE.
b) Press F5(DIST), F5(BINM), and F1(Bcd).
c) On the screen that appears, press F2(Var) to enter Variable
opposite Data.
d) Enter 4 opposite x, 12 opposite Numtrial, and .05 opposite
p.
e) Press F1(CALC) to and you will get 0.999...
f) Now, press MENU, highlight the RUN icon and press EXE to go to the RUN
screen.
g) Enter 1, - (minus sign),; then press SHIFT, Ans, EXE. to get 0.00223.....
Second Method if you want to know each probability:
Not complete yet!!!!
First I'll show a very easy way that gives only the answer; then I'll show a
method that takes more time, but
provides much more intermediate results.
Short Way:
a) Press 1, and then - , the subtraction sign.
b) Press 2ND, DISTR, move the cursor down to B:binomcdf( and press ENTER.
c) Enter numbers so that the display looks like this: binomcdf(12,
.05, 3.
d) Press ENTER and the answer, .0022364 will be displayed.
Longer Way:
a) Press 2ND, DISTR; then move the cursor to A:binompdf( and press ENTER.
b) Enter information so that your display looks like this:
binompdf(12, .05, {4, 5,6,7,8,9,10,11,12}). Be sure
to use braces rather than parentheses.
c) Press STO, 2ND, List 1 to tell the calculator which list to
store the individual values in.
Now, we want to also get the sum of all of these. Do that as follows:
d) Press ALPHA, : (the decimal point key); then 2ND, LIST, move the cursor to MATH, and press 5. The
expression
binompdf(12, .05, {4, 5,6,7,8,9,10,11,12}) : sum( should now be displayed on the home screen.
e) Press 2ND, List 1,. You should now have this expression:
binompdf(12, .05, {4, 5,6,7,8,9,10,11,12}) sum( List 1).
f) Press ENTER, and the answer, .0022364, will
be displayed. If you need the individual numbers,
they are in list List 1. Just press STAT, ENTER to see them.
Ex 2: Suppose in the above example you want to know the probability of
3 and fewer.
a) Press 2ND, DISTR, move the cursor down to B:binomcdf(
and press ENTER.
b) Enter numbers so that the display looks like this:
binomcdf(12, .05, 3.
c) Press ENTER and the answer, .997763... will be
displayed.
Ex 3: Suppose that, on average, one out of ten apples in a fruit
stand is unacceptable. What is the probability that
8, 9, or 10 of a set of 11 such apples are acceptable?
a) Press 2ND, LIST; move the cursor to MATH and press 5 to paste sum( to
the home screen.
b) Press 2ND, DISTR, ALPHA, A. You will now have sum(binomialPdf(
posted to the home screen.
c) Enter data so that you have sum(binomialPdf(11, .9, {8,9,10})) on
the home screen. Be sure to use braces
rather than parentheses enclosing the numbers 8, 9, 10.
d) Press ENTER and .667...will be displayed.
VIII. Hypothesis Testing:
1. Testing for Mean and z
Distribution with Data:
a) Enter
the data into List 1 or whatever list you choose.
b) Press STAT
and move the cursor over to TESTS.
c) Press 1 or
ENTER for Z-Test.
d) Move the
cursor to Data and press ENTER.
e) Opposite
µo, enter the mean for the null hypothesis.
f)
Opposite σ, if you are using the sample standard deviation and it is not given,
do the following: Press 2nd,
LIST, move the cursor to CALC and press 7. stdDev(, will now be displayed
opposite σ. Now, enter you
list number where the dats is stored by pressing 2nd, and the list number, for
example List 1 .
g) Enter List
1
opposite List and 1 opposite Freq.
h) Select the
proper condition for the alternative hypothesis.
i) Move
the cursor to Calculate and press ENTER.
j) If
you want to use the calculator to find the z-value or critical value, see those
procedures below.
2.
Testing for Mean and z Distribution with Statistics:
a) Press STAT
and move the cursor over to TESTS.
b) Press 1 or
ENTER for Z-Test.
c) Move the
cursor to Stats and press ENTER.
d) Opposite
µo, 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 2nd,
DISTR, 3 for invNorm(.
b) Enter
α for a left-tailed or 1-α for a right-tailed and press ENTER.
c) 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, DISTR, 3, and invNorm( will be pasted to the home screen.
b)
Enter numbers so that your entry looks like this: invNorm(.05, 5.25, .6.
For a left tail, enter the value
for α and for a right tail enter 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 List 1 or whatever list you choose.
b) Press STAT
and move the cursor over to TESTS.
c) Press 2 for T-Test.
d) Move the
cursor to Data and press ENTER.
e) Opposite µo, enter the mean for the null hypothesis.
f) Enter List
1
opposite List and 1 opposite Freq.
g) Select the
proper condition for the alternative hypothesis.
h) Move
the cursor to Calculate and press ENTER.
i) If
you are working a problem using the 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. There are different options
depending on your needs and whether
you're using a TI-83 Plus or a TI-84 Silver Edition. See "invT:
Finding a t-value Given α
and df:"
in section VII of
this document for the details of these options.
6.
Testing for Mean and T Distribution with Statistics:
a) Press STAT
and move the cursor over to TESTS.
b) Press 2 or
ENTER for T-Test.
c) Move the
cursor to Stat and press ENTER.
d) Opposite
µo, 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, LIST, cursor to MATH, and press 3; then
press ENTER. Enter List 1 and
press ENTER.
h) Select the
proper condition for the alternative hypothesis.
i) Move
the cursor to Calculate and press ENTER.
j) If
you are working a problem using the p-value test, read the p-value and compare
it with α or α-1 as appropriate.
k) 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. There are different options depending
on your needs and whether
you're using a TI-83 Plus or a TI-84 Silver Edition. See "invT:
Finding a t-value Given α
and df:"
in section VII of
this document
for the details of these options.
IX.
Simple Program for Calculating InverseT:
This is a simple program for those who
want to find t-values with a calculator. Because the TI-83Plus
has a fairly slow clock speed, a solution may take 20 seconds or so. When
you enter the program,, you can add more letters to the menu items if you
prefer. I have abbreviated them to save memory space in my calculator.
Using the Program:
a) After you’ve entered the
program, highlight the program name and press ENTER.
b) The program will ask for the confidence level, α, and then the
degrees of freedom, df. For this program,
α
is not divided by 2 when doing a two-tailed test. Remember that for a
c) You will then be presented
with a menu to select either right-tail, left-tail, or 2-tail. Select the one
appropriate by
either pressing the appropriate number or highlighting the number and pressing ENTER. The answer will be
displayed in
approximately 20 seconds.
PROGRAM:
: ”FKIZER 91906”
: INPUT “DF=”, D
: Menu(“SELECT”, Lft TL”, 1, “RT TL”, 2, “2-TL”, 3)
:
Lbl 1
: solve(tcdf(-1E9, X, D) – A, X, -1.7) →T
: Goto 4
: Lbl 2
: solve(tcdf(-1E9, X, D) –(1- A), X, 1.7) →T
: GoTo 4
: Lbl 3
: solve(tcdf(-1E9, X, D) – A/2, X, 1.7) →T
: Disp abs(T
:Lbl 4
:Disp T
X.
Statistics of two Populations:
1. Confidence Interval for Two
Dependent Populations:
Enter the data
from population 1 into List 1 and the data from population 2 into List
2.
Do this as follows:
a) Press STAT, ENTER, and enter the data in the
displayed lists.
b) After entering the data, press 2nd, QUIT to go to the home screen.
Now, store the paired differences in list L3 as follows:
c) From the home screen, press 2nd, List 1, minus sign, 2nd,
List 2.
d) Press STO, 2nd, L3. You should now have
List 1 - List 2 → L3 on the home
screen.
Now, find the confidence level as follows:
e) Press STAT, move the cursor to TESTS, and
press 8 for TInterval.
f) On the screen that appears, move
the cursor to "Data" and press ENTER; then enter 1 opposite Freq
and press ENTER.
g) Enter the confidence level you want opposite C-Level, for example .95.
h) Move the cursor down to “Calculate” and press ENTER. The confidence
interval and other statistics will be
displayed.
2. Confidence Interval for Two
Dependent Populations (Stats):
If you do not have data, but have the mean,
standard deviation, and n, use this procedure.
a) Press STAT, move the cursor to TESTS, and press 8
for TInterval.
b) On the screen that appears, move
the cursor to "Stats" and press ENTER.
c) Enter the sample mean, standard deviation, and
the number of data points opposite "n.".
d) Enter the confidence level you want opposite C-Level, for example .95.
f) Move the cursor down to “Calculate” and press ENTER. The confidence
interval and other statistics will be
displayed.
3. Confidence Interval for Two
Independent Populations (Stats):
a) Press STAT, move the cursor to
TESTS, and press 0 (zero).
b) On the screen that appears, move
the cursor to Stats and press ENTER.
c) Enter the sample means, standard
deviations, and number of data points, n, for each sample.
d) Set the confidence level you choose
opposite "C-Level."
e) Highlight "No" opposite "Pooled" if
there are no assumptions about the variations.
f) Move the cursor to "Calculate" and
press ENTER. The confidence interval along with other statistics will be
displayed.
4. Confidence Interval for Two
Independent Populations (Data):
Enter
the data from population 1 into List 1 and the data from population 2
into List 2.
Do this as follows:
a) Press STAT, ENTER, and enter the data in the
displayed lists.
b) After entering the data, press 2nd, QUIT to go to the home screen.
To go to the confidence interval screen do this:
c) Press STAT, move the cursor to
TESTS, and press 0 (zero).
d) On the screen that appears, move
the cursor to Data and press ENTER.
f) Opposite "List 1," press 2nd, List
1 and opposite "List2," press 2nd, List 2.
g) Set the confidence level you choose
opposite "C-Level."
h) Highlight "No" opposite "Pooled" if
there are no assumptions about the variations.
i) Move the cursor to "Calculate" and
press ENTER. The confidence interval along with other statistics will be
displayed.
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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may make single copies of this document for your own personal use and for the
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