Casio STAT 2 User manual

STAT 2
(Advanced Statistics Application)
Statistical Calculation (STAT) Soft-
ware for the ALGEBRA FX2.0
1. Modifications Made to ALGEBRA 2.0 by STAT2
2. Tests
3. Confidence Interval
4. Distribution

2
1.Modifications Made to ALGEBRA 2.0 by STAT2
uu
uu
uChanges to the Function Menu
Installing STAT2 changes the function menu of the STAT Mode list input screen (initial
screen) as shown below.
Pressing a function key that corresponds to the added item displays a menu that lets you
select one of the functions listed below.
•3(TEST) ... Test (Chapter 2, page 6)
•4(INTR) ... Confidence interval (Chapter 3, page 31)
•5(DIST) ... Distribution (Chapter 4, page 42)
The SORT and JUMP functions available with ALGEBRA FX2.0 are moved to the TOOL
menu (6and 1) by STAT2.
uu
uu
uCalculation of the Coefficient of Determination (r2) and MSE
STAT2 adds calculation of the coefficient of determination (r2) for quadratic regression, cubic
regression, and quartic regression. The following types of MSE calculations are also added
for each type of regression.
• Linear Regression ...
MSE = !
1
n–2
i=1
n
(y i –(ax i +b))
2
•Quadratic Regression ...
MSE = !
1
n–3i=1
n
(yi–(axi +bxi+c))2
2
•Cubic Regression ...
MSE = !
1
n–4i=1
n
(yi–(axi3+bxi +cxi+d ))2
2
•Quartic Regression ...
MSE = !
1
n–5
i=1
n
(y i –(ax i4 +bx i3 +cx i +dx i +e))
2
2
•Logarithmic Regression ...
MSE = !
1
n–2i=1
n
(yi–(a +b ln xi ))2

3
•Exponential Repression ...
MSE = !
1
n–2i=1
n
(ln yi–(ln a +bxi ))2
•Power Regression ...
MSE = !
1
n–2
i=1
n
(ln y i –(ln a +b ln x i ))
2
•Sin Regression ...
MSE = !
1
n–2
i=1
n
(y i –(a sin (bx i +c)+d ))
2
•Logistic Regression ...
MSE =
!
1
n–21+ ae
-bx
i
C
i=1
n
y i –
2
uu
uu
uEstimated Value Calculation for Regression Graphs
STAT2 adds a Y-CAL function that uses regression to calculate the estimated y-value for a
particular x-value after a paired-variable statistic regression graph is drawn.
The following is the general procedure for using the Y-CAL function.
1. After drawing a regression graph, press 62 (Y-CAL) to enter the graph selection mode,
and then press w.
If there are multiple graphs on the display, use fand cto select the graph you want, and
then press w.
•This causes an x-value input dialog box to appear.
2. Input the value you want for x and then press w.
•This causes the coordinates for x and y to appear at the bottom of the display, and moves
the pointer to the corresponding point on the graph.
3. Pressing vor a number key at this time causes the x-value input dialog box to reappear
so you can perform another estimated value calculation if you want.
4. After you are finished, press ito clear the coordinate values and the pointer from the
display.
·The pointer does not appear if the calculated coordinates are not within the display range.

4
·The coordinates do not appear if [Off] is specified for the [Coord] item of the [SETUP] screen.
·The Y-CAL function can also be used with a graph drawn by using DefG feature.
uu
uu
uRegression Formula Copy Function from a Regression Calculation Result
Screen
In addition to the existing regression formula copy function that lets you copy the regression
calculation result screen after drawing a statistical graph (such as Scatter Plot), STAT2 also
adds a function that lets you copy the regression formula obtained as the result of a regression
calculation. This type of copy operation is performed by pressing 6(COPY).
kk
kk
kTests, Confidence Interval, and Distribution Calculations
STAT2 adds functions for performing tests, confidence interval, and distribution calculations.
This manual fully describes each of these calculations in separate chapters: Chapter 2 Tests,
Chapter 3 Confidence Interval, and Chapter 4 Distribution.
uu
uu
uParameter Settings
The following describes the two methods you can use to make parameter settings for test,
confidence interval, and distribution calculations.
•Selection
With this method, you press the function key that corresponds to the setting you want to
select from the function menu.
•Value Input
With this method, you directly input the parameter value you want to input. In this case,
nothing appears in the function menu.
·Pressing ireturns to the list input screen, with the cursor in the same position it was at
before you started the parameter setting procedure.
·Pressing !i(QUIT) returns to the top of list input screen.
·Pressing wwithout pressing 1(CALC) under “Execute”item advances to calculation ex-
ecution. To return to the parameter setting screen, press i, A, or w.
uu
uu
uCommon Functions
•The symbol “!”appears in the upper right corner of the screen while execution of a calcula-
tion is being performed and while a graph is being drawn. Pressing Aduring this time
terminates the ongoing calculation or draw operation (AC Break).
•Pressing ior wwhile a calculation result or graph is on the display returns to the param-
eter setting screen. Pressing !i(QUIT) returns to the top of list input screen.

5
·Pressing Awhile a calculation result is on the display returns to the parameter setting screen.
•Pressing u5(G"T) after drawing a graph switches to the parameter setting screen
(G"T function). Pressing u5(G"T) again returns to the graph screen.
·The G"T function is disabled whenever you change a setting on the parameter setting screen
, or when you perform a u3(SET UP) or !K(V-Window) operation.
•You can perform the PICT menu's screen save or recall functions after drawing a graph.
·The ZOOM Function and SKETCH function are disabled.
The TRACE function is desabled, except for the geaph display of two-way ANOVA.
The graph screen cannot be scrolled.
•After drawing a graph, you can use a Save Result feature to save calculation results to a
specific list. Basically, all items are saved as they are displayed, except for the first line title.
·Each time you execute Save Result, any existing data in the list is replaced by the new
results.

6
2.Tests (TEST)
The ZTest provides a variety of different standardization-based tests. They make it possible
to test whether or not a sample accurately represents the population when the standard
deviation of a population (such as the entire population of a country) is known from previous
tests. Ztesting is used for market research and public opinion research that need to be
performed repeatedly.
1-Sample ZTest tests for unknown population mean when the population standard
deviation is known.
2-Sample ZTest tests the equality of the means of two populations based on independent
samples when both population standard deviations are known.
1-Prop ZTest tests for an unknown proportion of successes.
2-Prop ZTest tests to compare the propotion of successes from two populations.
The tTest tests the hypothesis when the population standard deviation is unknown. The
hypothesis that is the opposite of the hypothesis being proven is called the null hypothesis ,
while the hypothesis being proved is called the alternative hypothesis . The t-test is normally
applied to test the null hypothesis. Then a determination is made whether the null hypothesis
or alternative hypothesis will be adopted.
1-Sample tTest tests the hypothesis for a single unknown population mean when the
population standard deviation is unknown.
2-Sample tTest compares the population means when standard deviations are unknown.
Linear Reg tTest calculates the strength of the linear association of paired data.
!
2Test tests hypothesis concerning the proportion of samples included in each of a number
of independent groups. Mainly, it generates cross-tabulation of two categorical variables
(such as yes, no) and evaluates the independence of these variables. It could be used, for
example, to evaluate the relationship between whether or not a driver has ever been
involved in a traffic accident and that person’s knowledge of traffic regulations.
2-Sample FTest tests the hypothesis for the ratio of sample variances. It could be used, for
example, to test the carcinogenic effects of multiple suspected factors such as tobacco use,
alcohol, vitamin deficiency, high coffee intake, inactivity, poor living habits, etc.
ANOVA tests the hypothesis that the population means of the samples are equal when there
are multiple samples. It could be used, for example, to test whether or not different combina-
tions of materials have an effect on the quality and life of a final product.
One-Way ANOVA is used when there is one independent variable and one dependent
variable.
Two-Way ANOVA is used when there here are two independent variables and one depend-
ent variable.
The following pages explain various statistical calculation methods based on the principles
described above. Full details concerning statistical principles and terminology can be found
in any standard general statistics textbook.

7
On the initial STAT2 Mode screen, press 3(TEST) to display the test menu, which
contains the following items.
•3(TEST)b(Z) ... ZTests (p.7)
c(T) ... tTests (p.15)
d(!2) ... !2Test (p.23)
e(F) ... 2-Sample FTest (p.25)
f(ANOVA) ... ANOVA (p.27)
kk
kk
kZTests
uu
uu
uZTest Common Functions
You can use the following graph analysis functions after drawing a graph.
•1(Z) ... Displays zscore.
Pressing 1(Z) displays the zscore at the bottom of the display, and displays the pointer at
the corresponding location in the graph (unless the location is off the graph screen).
Two points are displayed in the case of a two-tail test. Use dand eto move the pointer.
Press ito clear the zscore.
•2(P) ... Displays p-value.
Pressing 2(P) displays the p-value at the bottom of the display without displaying the
pointer.
Press ito clear the p-value.
uu
uu
u1-Sample ZTest
This test is used when the sample standard deviation for a population is known to test the
hypothesis. The 1-Sample ZTest is applied to normal distribution.
Z= o–
0
"
µ
n
o: mean of sample
µ o: assumed population mean
" : population standard deviation
n: size of sample
# The following V-Window settings are used for
drawing the graph. Xmin = –3.2, Xmax = 3.2,
Xscale = 1, Ymin = –0.1, Ymax = 0.45, Yscale
=0.1
# Executing an analysis function automatically
stores the z and p values in alpha variables Z
and P, respectively.

8
Perform the following key operation from the statistical data list.
3(TEST)
b(Z)
b(1-Smpl)
The following shows the meaning of each item in the case of list data specification.
Data ............................ data type
µ .................................. population mean value test conditions (“G µ 0” specifies
two-tail test, “< µ 0” specifies lower one-tail test, “> µ 0”
specifies upper one-tail test.)
µ 0................................. assumed population mean
! .................................. population standard deviation ( ! > 0)
List .............................. list whose contents you want to use as data (List 1 to 20)
Freq ............................. frequency (1 or List 1 to 20)
Save Res .................... list for storage of calculation results (None or List 1 to 20)
Execute ....................... executes a calculation or draws a graph
The following shows the meaning of parameter data specification items that are different
from list data specification.
o.................................. mean of sample
n.................................. size of sample (positive integer)
After setting all the parameters, align the cursor with [Execute] and then press one of the
function keys shown below to perform the calculation or draw the graph.

9
•1(CALC) ... Performs the calculation.
•6(DRAW) ... Draws the graph.
Calculation Result Output Example
µ G11.4 ........................ direction of test
z.................................. Zscore
p.................................. p-value
o.................................. mean of sample
x " n-1 ............................. sample standard deviation
(Displayed only for Data: List setting)
n.................................. size of sample
# [Save Res] does not save the µ condition in
line 2.

10
uu
uu
u2-Sample ZTest
This test is used when the sample standard deviations for two populations are known to test
the hypothesis. The 2-Sample ZTest is applied to normal distribution.
Z= o
1
–o
2
"
n
1
1
2
"
n
2
2
2
+
o1: mean of sample 1
o2: mean of sample 2
" 1: population standard deviation of sample 1
" 2: population standard deviation of sample 2
n1: size of sample 1
n2: size of sample 2
Perform the following key operation from the statistical data list.
3(TEST)
b(Z)
c(2-Smpl)
The following shows the meaning of each item in the case of list data specification.
Data ............................ data type
µ 1................................. population mean value test conditions (“G µ 2”specifies two-
tail test, “< µ 2”specifies one-tail test where sample 1 is
smaller than sample 2, “> µ 2”specifies one-tail test where
sample 1 is greater than sample 2.)
" 1................................. population standard deviation of sample 1 ( " 1> 0)
" 2................................. population standard deviation of sample 2 ( " 2> 0)
List(1) .......................... list whose contents you want to use as sample 1 data
List(2) .......................... list whose contents you want to use as sample 2 data
Freq(1) ........................ frequency of sample 1 (positive integer)
Freq(2) ........................ frequency of sample 2 (positive integer)
Save Res .................... list for storage of calculation results (None or List 1 to 20)
Execute ....................... executes a calculation or draws a graph
The following shows the meaning of parameter data specification items that are different
from list data specification.

11
o1................................. mean of sample 1
n1................................. size (positive integer) of sample 1
o2................................. mean of sample 2
n2................................. size (positive integer) of sample 2
After setting all the parameters, align the cursor with [Execute] and then press one of the
function keys shown below to perform the calculation or draw the graph.
•1(CALC) ... Performs the calculation.
•6(DRAW) ... Draws the graph.
Calculation Result Output Example
µ 1G µ 2........................... direction of test
z ................................... Z score
p.................................. p-value
o1................................. mean of sample 1
o2................................. mean of sample 2
x1 " n-1 ............................ standard deviation of sample 1
(Displayed only for Data: List Setting)
x2 " n-1 ............................ standard deviation of sample 2
(Displayed only for Data: List Setting.)
n1................................. size of sample 1
n2................................. size of sample 2
# [Save Res] does not save the µ 1condition in
line 2.

12
uu
uu
u1-Prop ZTest
This test is used to test for an unknown proportion of successes. The 1-Prop ZTest is
applied to normal distribution.
Z= n
x
n
p
0
(1–p
0
)
–p
0
p0: expected sample proportion
n: size of sample
Perform the following key operation from the statistical data list.
3(TEST)
b(Z)
d(1-Prop)
Prop ............................ sample proportion test conditions (“Gp0”specifies two-tail
test, “<p0”specifies lower one-tail test, “>p0”specifies upper
one-tail test.)
p0................................. expected sample proportion (0 < p0< 1)
x.................................. sample value (x>0 integer)
n.................................. size of sample (positive integer)
Save Res .................... list for storage of calculation results (None or List 1 to 20)
Execute ....................... executes a calculation or draws a graph
After setting all the parameters, align the cursor with [Execute] and then press one of the
function keys shown below to perform the calculation or draw the graph.
•1(CALC) ... Performs the calculation.
•6(DRAW) ... Draws the graph.
Calculation Result Output Example
PropG0.5 .................... direction of test
z ................................... Z score
p .................................. p-value
ˆp .................................. estimated sample proportion
n .................................. size of sample
# [Save Res] does not save the Prop condition
in line 2.

13
uu
uu
u2-Prop ZTest
This test is used to compare the proportion of successes. The 2-Prop ZTest is applied to
normal distribution.
Z= n1
x1
n2
x2
–
p(1–p)n1
1
n2
1
+
x1: data value of sample 1
x2: data value of sample 2
n1: size of sample 1
n2: size of sample 2
ˆp: estimated sample proportion
Perform the following key operation from the statistical data list.
3(TEST)
b(Z)
e(2-Prop)
p1................................. sample proportion test conditions (“Gp2”specifies two-tail
test, “<p2”specifies one-tail test where sample 1 is less than
sample 2, “>p2”specifies upper one-tail test where sample 1
is greater than sample 2.)
x1................................. data value (x1>0 integer) of sample 1
n1................................. size (positive integer) of sample 2
x2................................. data value (x2>0 integer) of sample 1
n2................................. size (positive integer) of sample 2
Save Res .................... list for storage of calculation results (None or List 1 to 20)
Execute ....................... executes a calculation or draws a graph
After setting all the parameters, align the cursor with [Execute] and then press one of the
function keys shown below to perform the calculation or draw the graph.
•1(CALC) ... Performs the calculation.
•6(DRAW) ... Draws the graph.
Calculation Result Output Example

14
p1>p2............................ direction of test
z.................................. Zscore
p.................................. p-value
ˆp1................................. estimated proportion of sample 1
ˆp2................................. estimated proportion of sample 2
ˆp.................................. estimated sample proportion
n1................................. size of sample 1
n2................................. size of sample 2
# [Save Res] does not save the p1condition in
line 2.

15
kk
kk
ktTests
uu
uu
utTest Common Functions
You can use the following graph analysis functions after drawing a graph.
•1(T) ... Displays tscore.
Pressing 1(T) displays the tscore at the bottom of the display, and displays the pointer at the
corresponding location in the graph (unless the location is off the graph screen).
Two points are displayed in the case of a two-tail test. Use dand eto move the pointer.
Press ito clear the tscore.
•2(P) ... Displays p-value.
Pressing 2(P) displays the p-value at the bottom of the display without displaying the pointer.
Press ito clear the p-value.
# The following V-Window settings are used for
drawing the graph. Xmin = –3.2, Xmax = 3.2,
Xscale = 1, Ymin = –0.1, Ymax = 0.45, Yscale
=0.1
# Executing an analysis function automatically
stores the t and p values in alpha variables T
and P, respectively.

16
uu
uu
u1-Sample tTest
This test uses the hypothesis test for a single unknown population mean when the popula-
tion standard deviation is unknown. The 1-Sample tTest is applied to t-distribution.
t= o–0
µ
"
x n–1
n
o: mean of sample
µ 0: assumed population mean
x " n-1 : sample standard deviation
n: size of sample
Perform the following key operation from the statistical data list.
3(TEST)
c(T)
b(1-Smpl)
The following shows the meaning of each item in the case of list data specification.
Data ............................ data type
µ .................................. population mean value test conditions (“G µ 0”specifies two-
tail test, “< µ 0”specifies lower one-tail test, “> µ 0”specifies
upper one-tail test.)
µ 0................................. assumed population mean
List .............................. list whose contents you want to use as data
Freq ............................. frequency
Save Res .................... list for storage of calculation results (None or List 1 to 20)
Execute ....................... executes a calculation or draws a graph
The following shows the meaning of parameter data specification items that are different
from list data specification.
o.................................. mean of sample
x " n-1 ............................. sample standard deviation (x " n-1 > 0)
n.................................. size of sample (positive integer)
After setting all the parameters, align the cursor with [Execute] and then press one of the
function keys shown below to perform the calculation or draw the graph.
•1(CALC) ... Performs the calculation.
•6(DRAW) ... Draws the graph.

17
Calculation Result Output Example
µ G11.3 ...................... direction of test
t ................................... tscore
p.................................. p-value
o.................................. mean of sample
x " n-1 ............................. Sample standard deviation
n.................................. size of sample
# [Save Res] does not save the µ condition in
line 2.

18
uu
uu
u2-Sample tTest
2-Sample tTest compares the population means when standard deviations are unknown.
The 2-Sample tTest is applied to t-distribution.
t= o
1
–o
2
x
1n–1
2
"
n
1
+x
2n–1
2
"
n
2
o1: mean of sample 1
o2: mean of sample 2
x1 " n-1 : standard deviation of sample 1
x2 " n-1 : standard deviation of sample 2
n1: size of sample 1
n2: size of sample 2
This formula is applicable when the sample is not pooled, and the denominator is different
when the sample is pooled.
Degrees of freedom df and xp " n-1 differs according to whether or not pooling is in effect.
The following applies when pooling is in effect.
df = n1+ n2–2
x
p n–1
= "
n
1
+n
2
–2
(n
1
–1)x
1n–12
+(n
2
–1)x
2n–12
""
The following applies when pooling is not in effect.
df = 1
C
2
n
1
–1+(1–C )
2
n
2
–1
C=
x
1n–12
"
n
1
+x
2n–12
"
n
2
x
1n–12
"
n
1
Perform the following key operation from the statistical data list.
3(TEST)
c(T)
c(2-Smpl)

19
The following shows the meaning of each item in the case of list data specification.
Data ............................ data type
µ 1................................. sample mean value test conditions (“G µ 2”specifies two-tail
test, “< µ 2”specifies one-tail test where sample 1 is smaller
than sample 2, “> µ 2”specifies one-tail test where sample 1 is
greater than sample 2.)
List(1) .......................... list whose contents you want to use as data of sample 1
List(2) .......................... list whose contents you want to use as data of sample 2
Freq(1) ........................ frequency of sample 1 (positive integer)
Freq(2) ........................ frequency of sample 2 (positive integer)
Pooled ......................... pooling On (in effect) or Off (not in effect)
Save Res .................... list for storage of calculation results (None or List 1 to 20)
Execute ....................... executes a calculation or draws a graph
The following shows the meaning of parameter data specification items that are different
from list data specification.
o1................................. mean of sample 1
x1 " n-1 ............................ standard deviation (x1 " n-1 > 0) of sample 1
n1................................. size (positive integer) of sample 2
o2................................. mean of sample 2
x2 " n-1 ............................ standard deviation (x2 " n-1 > 0) of sample 2
n2................................. size (positive integer) of sample 2
After setting all the parameters, align the cursor with [Execute] and then press one of the
function keys shown below to perform the calculation or draw the graph.
•1(CALC) ... Performs the calculation.
•6(DRAW) ... Draws the graph.
Calculation Result Output Example
µ 1G µ 2........................... direction of test
t ................................... tscore

20
p.................................. p-value
df ................................. degrees of freedom
o1................................. mean of sample 1
o2................................. mean of sample 2
x1 " n-1 ............................ standard deviation of sample 1
x2 " n-1 ............................ standard deviation of sample 2
xp " n-1 ............................ pooled sample standard deviation (Displayed only when
Pooled: On Setting.)
n1................................. size of sample 1
n2................................. size of sample 2
# [Save Res] does not save the µ 1 condition in
line 2.