Casio STAT 2 User Manual

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STAT 2 (Advanced Statistics Application)

Statistical Calculation (STAT) Software for the ALGEBRA FX2.0

1. Modifications Made to ALGEBRA 2.0 by STAT2

2. Tests

3. Confidence Interval

4. Distribution

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1.Modifications Made to ALGEBRA 2.0 by STAT2

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 (6 and 1) by STAT2.

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 =

n – 2

i=1

(yi – (axi+ b))

• Quadratic Regression ...

MSE =

n – 3

i=1

(yi – (ax

+ bxi+ c))

• Cubic Regression ...

MSE =

n – 4

i=1

(yi – (ax

+ bx

+ cx

+d ))

• Quartic Regression ...

MSE =

n – 5

i=1

(yi – (ax

+ bx

+ cx

+ dx

+ e))

• Logarithmic Regression ...

MSE =

n – 2

i=1

(yi – (a + b ln xi ))

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• Exponential Repression ...

MSE =

n – 2

i=1

(ln yi – (ln a + bxi ))

• Power Regression ...

MSE =

n – 2

i=1

(ln yi – (ln a + b ln xi ))

• Sin Regression ...

MSE =

n – 2

i=1

(yi – (a sin (bxi + c) + d ))

• Logistic Regression ...

MSE =

n – 2 1 + ae

-bx

i=1

yi –

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 f and c to 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 v or 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 i to 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.

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· 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.

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).

k Tests, 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.

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 i returns 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 w without pressing 1 (CALC) under “Execute” item advances to calculation ex-

ecution. To return to the parameter setting screen, press i, A, or w.

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 A during this time terminates the ongoing calculation or draw operation (AC Break).

• Pressing i or w while 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.

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· Pressing A while a calculation result is on the display returns to the parameter setting screen.

• Pressing u 5 (G"T) after drawing a graph switches to the parameter setting screen

(G"T function). Pressing u 5 (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 u 3 (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.

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2.Tests (TEST)

The Z Test 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. Z testing is used for market research and public opinion research that need to be performed repeatedly.

1-Sample Z Test tests for unknown population mean when the population standard deviation is known.

2-Sample Z Test tests the equality of the means of two populations based on independent samples when both population standard deviations are known.

1-Prop Z Test tests for an unknown proportion of successes.

2-Prop Z Test tests to compare the propotion of successes from two populations.

The t Test 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 t Test tests the hypothesis for a single unknown population mean when the population standard deviation is unknown.

2-Sample t Test compares the population means when standard deviations are unknown.

Linear Reg t Test calculates the strength of the linear association of paired data.

Test 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 F Test 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 combinations 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 dependent 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.

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On the initial STAT2 Mode screen, press 3 (TEST) to display the test menu, which contains the following items.

• 3(TEST)b(Z) ... Z Tests (p.7)

c(T) ... t Tests (p.15)

d(!2) ... !2 Test (p.23)

e(F) ... 2-Sample F Test (p.25)

f(ANOVA) ... ANOVA (p.27)

k Z Tests

uZ Test Common Functions

You can use the following graph analysis functions after drawing a graph.

• 1(Z) ... Displays z score.

Pressing 1 (Z) displays the z score 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 d and e to move the pointer. Press i to clear the z score.

• 2(P) ... Displays p-value.

Pressing 2 (P) displays the p-value at the bottom of the display without displaying the pointer. Press i to clear the p-value.

u1-Sample Z Test

This test is used when the sample standard deviation for a population is known to test the hypothesis. The 1-Sample Z Test is applied to normal distribution.

Z =

o –

o : mean of sample

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.

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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

.................................. population mean value test conditions (“G

0” specifies

two-tail test, “<

0” specifies lower one-tail test, “> µ0”

specifies upper one-tail test.)

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.

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• 1(CALC) ... Performs the calculation.

• 6(DRAW) ... Draws the graph.

Calculation Result Output Example

G11.4

........................

direction of test

z .................................. Z score

p .................................. p-value

o .................................. mean of sample

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.

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u2-Sample

Z Test

This test is used when the sample standard deviations for two populations are known to test the hypothesis. The 2-Sample Z Test is applied to normal distribution.

Z =

o1 – o

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

1 ................................. population mean value test conditions (“G µ2” specifies two-

tail test, “<

2” specifies one-tail test where sample 1 is

smaller than sample 2, “>

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.

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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

2 ........................... direction of test

z ................................... Z score

p .................................. p-value

o1 ................................. mean of sample 1

o2 ................................. mean of sample 2

n-1 ............................ standard deviation of sample 1

(Displayed only for Data: List Setting)

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

1 condition in

line 2.

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u1-Prop

Z Test

This test is used to test for an unknown proportion of successes. The 1-Prop Z Test is applied to normal distribution.

Z =

(1– p0)

– p

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 (“G p0” 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.

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u2-Prop Z Test

This test is used to compare the proportion of successes. The 2-Prop Z Test is applied to normal distribution.

Z =

–

p(1 – p )

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 (“G p2” 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

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p1>p2 ............................ direction of test

z .................................. Z score

p .................................. p-value

ˆp 1 ................................. estimated proportion of sample 1

ˆp 2 ................................. 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 p1 condition in

line 2.

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k t Tests

u t Test Common Functions

You can use the following graph analysis functions after drawing a graph.

• 1(T) ... Displays t score.

Pressing 1 (T) displays the t score 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 d and e to move the pointer.

Press i to clear the t score.

• 2(P) ... Displays p-value.

Pressing 2 (P) displays the p-value at the bottom of the display without displaying the pointer.

Press i to 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.

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u1-Sample t Test

This test uses the hypothesis test for a single unknown population mean when the population standard deviation is unknown. The 1-Sample t Test is applied to t-distribution.

t =

o –

n–1

o : mean of sample

0 : assumed population mean

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

.................................. population mean value test conditions (“G

0” specifies two-

tail test, “<

0” specifies lower one-tail test, “> µ0” specifies

upper one-tail test.)

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

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.

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Calculation Result Output Example

G 11.3 ...................... direction of test

...................................

t score

p .................................. p-value

o .................................. mean of sample

n-1 ............................. Sample standard deviation

n .................................. size of sample

# [Save Res] does not save the µ condition in

line 2.

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u2-Sample t Test

2-Sample t Test compares the population means when standard deviations are unknown. The 2-Sample t Test is applied to t-distribution.

t =

o1 – o

1 n–1

2 n–1

o1 : mean of sample 1 o2 : mean of sample 2

n-1 : standard deviation of sample 1

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.

= n1 + n2 – 2

p n–1

+ n

– 2

–1)x

n–1

+(n2–1)x

n–1

The following applies when pooling is not in effect.

df =

n1–1

(1–C )

n2–1

C =

1 n–1

2 n–1

1 n–1

Perform the following key operation from the statistical data list.

3(TEST)

c(T)

c(2-Smpl)

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The following shows the meaning of each item in the case of list data specification.

Data ............................ data type

1 ................................. sample mean value test conditions (“G µ2” specifies two-tail

test, “<

2” specifies one-tail test where sample 1 is smaller

than sample 2, “>

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

n-1 ............................ standard deviation (x1"n-1 > 0) of sample 1

n1 ................................. size (positive integer) of sample 2

o2 ................................. mean of sample 2

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

µ1Gµ

2 ........................... direction of test

...................................

t score

Page 20

p .................................. p-value

df ................................. degrees of freedom

o1 ................................. mean of sample 1

o2 ................................. mean of sample 2

n-1 ............................ standard deviation of sample 1

n-1 ............................ standard deviation of sample 2

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

µ1

condition in

line 2.

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uLinearReg

t Test

LinearReg t Test treats paired-variable data sets as (x, y) pairs and plots all data on a graph. Next, a straight line (y = a + bx) is drawn through the area where the greatest number of plots are located and the degree to which a relationship exists is calculated.

b =

( x – o)( y – p)

i=1

(x – o)

i=1

a = p – bo t = r

n – 2

1 – r

a : intercept b : slope of the line n : size of sample (n>3) r : correlation coefficient r

: c

oefficient of determination

Perform the following key operation from the statistical data list.

3(TEST)

c(T)

d(LinReg)

The following shows the meaning of each item in the case of list data specification.

& %............................ p-value test conditions (“G 0” specifies two-tail test, “< 0”

specifies lower one-tail test, “> 0” specifies upper one-tail test.)

XList ............................ list for x-axis data

YList ............................ list for y-axis data

Freq ............................. frequency

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

# You cannot draw a graph for LinearReg t

Test.

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Calculation Result Output Example

G 0 &

G 0 .............. direction of test

t ................................... t score

p .................................. p-value

df ................................. degrees of freedom

a .................................. constant term

b .................................. coefficient

s .................................. standard error

r .................................. correlation coefficient

................................. coefficient of determination

Pressing 6 (COPY) while a calculation result is on the display copies the regression formula to the graph formula editor.

When there is a list specified for the [Resid List] item on the SET UP screen, regression formula residual data is automatically saved to the specified list after the calculation is finished.

# [Save Res] does not save the $ &

conditions in line 2.

# When the list specified by [Save Res] is the

same list specified by the [Resid List] item on the SET UP screen, only[Resid List] data is saved in the list.

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k !

Test

!2 Test sets up a number of independent groups and tests hypothesis related to the proportion of the sample included in each group. The !2 Test is applied to dichotomous variables (variable with two possible values, such as yes/no).

expected counts

Fij =

i=1

j=1

n : all data values

!2 =

i=1

(xij – Fij)

j=1

Perform the following key operation from the statistical data list.

3(TEST)

d(!2)

Next, specify the matrix that contains the data. The following shows the meaning of the above item.

Observed .................... name of matrix (A to Z) that contains observed counts (all cells

positive integers)

Expected ..................... name of matrix (A to Z) that is for saving expected frequency

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation or draws a graph

# The matrix must be at least two lines by two

columns. An error occurs if the matrix has only one line or one column.

# Pressing 2 ('MAT) while setting

parameters enters the MATRIX editor, which you can use to edit and view the contents of matrices.

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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

................................. !2 value

p .................................. p-value

df ................................. degrees of freedom

You can use the following graph analysis functions after drawing a graph.

• 1(CHI) ... Displays

value.

Pressing 1 (CHI) displays the

value 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).

Press i to clear the

value.

• 2(P) ... Displays p-value.

Pressing 2 (P) displays the p-value at the bottom of the display without displaying the pointer.

Press i to clear the p-value.

# Pressing 6 (('MAT) while a calculation

result is displayed enters the MATRIX editor, which you can use to edit and view the contents of matrices.

# The following V-Window settings are used for

drawing the graph.Xmin = 0, Xmax = 11.5, Xscale = 2, Ymin = –0.1, Ymax = 0.5, Yscale =0.1

# Executing an analysis function automatically

stores the

and p values in alpha variables

C and P, respectively.

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k 2-Sample F Test

2-Sample F Test tests the hypothesis for the ratio of sample variances. The F Test is applied to F distribution.

F =

1 n–1

2 n–1

Perform the following key operation from the statistical data list.

3(TEST)

e(F)

The following is the meaning of each item in the case of list data specification.

Data ............................ data type

1 ................................. population standard deviation test conditions (“G "2”

specifies two-tail test, “<

2” specifies one-tail test where

sample 1 is smaller than sample 2, “>

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

Freq(2) ........................ frequency 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

The following shows the meaning of parameter data specification items that are different from list data specification.

n-1 ............................ standard deviation (x1

n-1

0) of sample 1

n1 ................................. size (positive integer) of sample 1

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.

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Calculation Result Output Example

"1G"

2 .......................... direction of test

F .................................. F value

p .................................. p-value

o1 ................................. mean of sample 1 (Displayed only for Data: List Setting)

o2 ................................. mean of sample 2 (Displayed only for Data: List Setting)

n-1 ............................ standard deviation of sample 1

n-1 ............................ standard deviation of sample 2

n1 ................................. size of sample 1

n2 ................................. size of sample 2

You can use the following graph analysis functions after drawing a graph.

• 1(F) ... Displays F value.

Pressing 1 (F) displays the F value 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 d and e to move the pointer.

Press i to clear the F value.

• 2(P) ... Displays p-value.

Pressing 2 (P) displays the p-value at the bottom of the display without displaying the pointer.

Press i to clear the p-value.

# [Save Res] does not save the

1 condition in

line 2.

# V-Window settings are automatically

optimized for drawing the graph.

# Executing an analysis function automatically

stores the

F and p values in alpha variables

F and P, respectively

Page 27

k ANOVA

ANOVA tests the hypothesis that the population means of the samples are equal when there are multiple samples. 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 dependent variable. This Two-Way ANOVA calculation is available under the condition that is to prepare more than two experimental data as each dependent data.

Perform the following key operation from the statistical data list.

3(TEST)

f(ANOVA)

The following is the meaning of each item in the case of list data specification.

How Many ................... selects One-Way ANOVA or Two-Way ANOVA (number of lev-

els).

Factor A....................... category list.

Dependnt .................... list to be used for sample data.

Save Res .................... first list for storage of calculation results.

Execute ....................... executes a calculation or draws a graph (Two-Way ANOVA only)

The following item appears in the case of Two-Way ANOVA only.

Factor B ...................... category list.

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 (Two-Way ANOVA only).

Calculation results are displayed in table form, just as they appear in science books.

# [Save Res] saves each vertical column of the

table into its own list. The leftmost column is saved in the specified list, and each subsequent column to the right is saved in

the next sequentially numbered list. Up to five lists can be used for storing columns. You can specify an first list number in the range of 1 to 16.

Page 28

Calculation Result Output Example

One-Way ANOVA

Line 1 (A) .................... Factor A df value, SS value, MS value, F value, p-value

Line 2 (ERR) ............... Error df value, SS value, MS value

Two-Way ANOVA

Line 1 (A) .................... Factor A df value, SS value, MS value, F value, p-value

Line 2 (B) .................... Factor B df value, SS value, MS value, F value, p-value

Line 3 (AB) .................. Factor A x Factor B df value, SS value, MS value, F value,

p-value

Line 4 (ERR) ............... Error df value, SS value, MS value

F .................................. F value

p .................................. p-value

df ................................. degrees of freedom

SS ................................ sum of squares

MS ............................... mean squares

With Two-Way ANOVA, you can draw Interaction Plot graphs. The number of graphs depends on Factor B, while the number of X-axis data depends on the Factor A. The Y-axis is the average value of each category.

You can use the following graph analysis function after drawing a graph.

• 1(TRACE) ... Trace function

Pressing d or e moves the pointer on the graph in the corresponding direction. When there are multiple graphs, you can move between graphs by pressing f and c.

Press i to clear the pointer from the display.

# Graphing is available with Two-Way ANOVA

only. V-Window settings are performed automatically, regardless of SET UP screen settings.

# Using the TRACE function automatically

stores the number of conditions to alpha variable A and the mean value to variable M, respectively.

Page 29

k ANOVA (Two-Way)

uDescription

The nearby table shows measurement results for a metal product produced by a heat treatment process based on two treatment levels: time (A) and temperature (B). The experiments were repeated twice each under identical conditions.

Perform analysis of variance on the following null hypothesis, using a significance level of 5%.

Ho : No change in strength due to time Ho : No change in strength due to heat treatment temperature Ho : No change in strength due to interaction of time and heat treatment temperature

uSolution

Use two-way ANOVA to test the above hypothesis. Input the above data as shown below.

List1={1,1,1,1,2,2,2,2} List2={1,1,2,2,1,1,2,2} List3={113,116,139,132,133,131,126,122 }

Define List 3 (the data for each group) as Dependent. Define List 1 and List 2 (the factor numbers for each data item in List 3) as Factor A and Factor B respectively. Executing the test produces the following results.

• Time differential (A) level of significance P = 0.2458019517 The level of significance (p = 0.2458019517) is greater than the significance level (0.05), so the hypothesis does not reject.

• Temperature differential (B) level of significance P = 0.04222398836 The level of significance (p = 0.04222398836) is less than the significance level (0.05), so the hypothesis rejects.

• Interaction (A & B) level of significance P = 2.78169946e-3 The level of significance (p = 2.78169946e-3) is less than the significance level (0.05), so the hypothesis rejects.

The above test indicates that the time differential is not significant, the temperature differential is significant, and interaction is highly significant.

B (Heat Treatment Temperature) B1 B2

A1 113 , 116

133 , 131

139 , 132

126 , 122

A (Time)

Page 30

uInput Example

uResults

Page 31

3. Confidence Interval (INTR)

A confidence interval is a range (interval) that includes the population mean value.

A confidence interval that is too broad makes it difficult to get an idea of where the population value (true value) is located. A narrow confidence interval, on the other hand, limits the population value and makes it possible to obtain reliable results. The most commonly used confidence levels are 95% and 99%. Raising the confidence level broadens the confidence interval, while lowering the confidence level narrows the confidence level, but it also increases the chance of accidently overlooking the population value. With a 95% confidence interval, for example, the population value is not included within the resulting intervals 5% of the time.

When you plan to conduct a survey and then

t test and Z test the data, you must also

consider the sample size, confidence interval width, and confidence level. The confidence level changes in accordance with the application.

1-Sample Z Interval calculates the confidence interval for an unknown population mean when standard deviation is known.

2-Sample Z Interval calculates the confidence interval for the difference between two population means when the standard deviations of two samples are known.

1-Prop Z Interval uses the number of data to calculate the confidence interval for an unknown proportion of successes .

2-Prop Z Interval uses the number of data items to calculate the confidence interval for the difference between the propotion of successes in two populations .

1-Sample t Interval calculates the confidence interval for an unknown population mean when the population standard deviation is unknown .

2-Sample t Interval calculates the confidence interval for the difference between two population means when both population standard deviations are unknown.

On the initial STAT Mode screen, press 4 (INTR) to display the confidence interval menu, which contains the following items.

• 4(INTR)b(Z) ... Z intervals (p.33)

c(T)... t intervals (p.38)

# There is no graphing for confidence interval

functions.

Page 32

uGeneral Confidence Interval Precautions

Inputting a value in the range of 0 < C-Level<1 for the C-Level setting sets you value you input. Inputting a value in the range of 1 < C-Level<100 sets a value equivalent to your input divided by 100.

# Inputting a value of 100 or greater, or a

negative value causes an error (Ma ERROR).

Page 33

k Z Interval

u1-Sample Z Interval

1-Sample Z Interval calculates the confidence interval for an unknown population mean when standard deviation is known.

The following is the confidence interval.

Left = o – Z

Right = o + Z

However, ! is not the confidence level itself. The value 100 (1-!) % is the confidence level.

When the confidence level is 95%, for example, inputting 0.95 produces 1 – 0.95 = 0.05 = !.

Perform the following key operation from the statistical data list.

4(INTR)

b(Z)

b(1-Smpl)

The following shows the meaning of each item in the case of list data specification.

Data ............................ data type

C-Level ........................ confidence level (0 < C-Level < 1)

.................................. population standard deviation (" > 0)

List .............................. list whose contents you want to use as sample data

Freq ............................. sample frequency

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

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)

Page 34

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

Left .............................. interval lower limit (left edge)

Right ............................ interval upper limit (right edge)

o .................................. mean of sample

n-1 ............................. sample standard deviation

(Displayed only for Data: List Setting)

n .................................. size of sample

u 2-Sample Z Interval

2-Sample Z Interval calculates the confidence interval for the difference between two population means when the standard deviations of two samples are known. The following is confidence interval. The value 100 (1-!) % is the confidence level.

Left = (o1 – o2) – Z

Right = (o

– o2) + Z

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.

4(INTR)

b(Z)

c(2-Smpl)

Page 35

The following shows the meaning of each item in the case of list data specification.

Data ............................ data type

C-Level ........................ confidence level (0 < C-Level < 1)

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 data of sample 1

List(2) .......................... list whose contents you want to use as data of sample 2

Freq(1) ........................ frequency of sample 1

Freq(2) ........................ frequency of sample 2

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

The following shows the meaning of parameter data specification items that are different from list data specification.

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 the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

Left .............................. interval lower limit (left edge)

Right ............................ interval upper limit (right edge)

o1 ................................. mean of sample 1

o2 ................................. mean of sample 2

n-1 ............................ standard deviation of sample 1

(Displayed only for Data: List Setting)

n-1 ............................ standard deviation of sample 2

(Displayed only for Data: List Setting)

n1 ................................. size of sample 1

n2 ................................. size of sample 2

Page 36

u1-Prop Z Interval

1-Prop Z Interval uses the number of data to calculate the confidence interval for an unknown proportion of successes.

The following is the confidence interval. The value 100 (1-!) % is the confidence level.

Left = – Z

Right = + Z

x n

1–

x n

n1nxn

1–

n : size of sample x : data

Perform the following key operation from the statistical data list.

4(INTR)

b(Z)

d(1-Prop)

Data is specified using parameter specification. The following shows the meaning of each item.

C-Level ........................ confidence level (0 < C-Level < 1)

x .................................. data (0 or positive 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

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

Left .............................. interval lower limit (left edge)

Right ............................ interval upper limit (right edge)

ˆp .................................. estimated sample proportion

n .................................. size of sample

Page 37

u 2-Prop

Z Interval

2-Prop Z Interval uses the number of data items to calculate the confidence interval for the defference between the proportion of successes in two populations.

The following is the confidence interval. The value 100 (1-!) % is the confidence level.

Left = – – Z

1–

Right = – + Z

1–

n1, n2 : size of sample

x1, x2 : data

Perform the following key operation from the statistical data list.

4(INTR)

b(Z)

e(2-Prop)

Data is specified using parameter specification. The following shows the meaning of each item.

C-Level ........................ confidence level (0 < C-Level < 1)

x1 ................................. data value (x1 > 0) of sample 1

n1 ................................. size (positive integer) of sample 1

x2 ................................. data value (x2 > 0) of sample 2

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

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

Page 38

Left .............................. interval lower limit (left edge)

Right ............................ interval upper limit (right edge)

ˆp 1 ................................. estimated sample propotion for sample 1

ˆp 2 ................................. estimated sample propotion for sample 2

n1 ................................. size of sample 1

n2 ................................. size of sample 2

k t Interval

u 1-Sample t Interval

1-Sample t Interval calculates the confidence interval for an unknown population mean when the population standard deviation is unknown.

The following is the confidence interval. The value 100 (1-!) % is the confidence level.

Left = o– t

n – 1

Right = o+ t

n – 1

n–1

Perform the following key operation from the statistical data list.

4(INTR)

c(T)

b(1-Smpl)

The following shows the meaning of each item in the case of list data specification.

Data ............................ data type

C-Level ........................ confidence level (0 < C-Level < 1)

List .............................. list whose contents you want to use as sample data

Freq ............................. sample frequency

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

The following shows the meaning of parameter data specification items that are different from list data specification.

Page 39

o .................................. mean of sample

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 the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

Left .............................. interval lower limit (left edge)

Right ............................ interval upper limit (right edge)

o .................................. mean of sample

n-1 ............................. sample standard deviation

n .................................. size of sample

u 2-Sample t Interval

2-Sample t Interval calculates the confidence interval for the difference between two population means when both population standard deviations are unknown. The t Interval is applied to t distribution.

The following confidence interval applies when pooling is in effect. The value 100 (1-!) % is the confidence level.

Left = (o1 – o2)– t

Right = (o1 – o2)+ t

n1+n

–2

n–1

n1+n

–2

n–1

Page 40

The following confidence interval applies when pooling is not in effect. The value 100 (1-!) % is the confidence level.

Left = (o1 – o2)– t

Right = (o

– o2)+ t

1 n–1

2 n–1

1 n–1

2 n–1

C =

df =

n1–1

(1–C )

n2–1

1 n–1

2 n–1

Perform the following key operation from the statistical data list.

4(INTR)

c(T)

c(2-Smpl)

The following shows the meaning of each item in the case of list data specification.

Data ............................ data type

C-Level ........................ confidence level (0 < C-Level < 1)

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

Freq(2) ........................ frequency of sample 2

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

The following shows the meaning of parameter data specification items that are different from list data specification.

Page 41

o1 ................................. mean of sample 1

n-1 ............................ standard deviation (x1"n-1 > 0) of sample 1

n1 ................................. size (positive integer) of sample 1

o2 ................................. mean of sample 2

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 the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

Left .............................. interval lower limit (left edge)

Right ............................ interval upper limit (right edge)

df ................................. degrees of freedom

o1 ................................. mean of sample 1

o2 ................................. mean of sample 2

n-1 ............................ standard deviation of sample 1

n-1 ............................ standard deviation of sample 2

n-1 ............................ pooled sample standard deviation

(Displayed only when Pooled: On Setting.)

n1 ................................. size of sample 1

n2 ................................. size of sample 2

Page 42

4. Distribution (DIST)

There is a variety of different types of distribution, but the most well-known is “normal distribution,” which is essential for performing statistical calculations. Normal distribution is a symmetrical distribution centered on the greatest occurrences of mean data (highest frequency), with the frequency decreasing as you move away from the center. Poisson distribution, geometric distribution, and various other distribution shapes are also used, depending on the data type.

Certain trends can be determined once the distribution shape is determined. You can calculate the probability of data taken from a distribution being less than a specific value.

For example, distribution can be used to calculate the yield rate when manufacturing some product. Once a value is established as the criteria, you can calculate normal probability when estimating what percent of the products meet the criteria. Conversely, a success rate target (80% for example) is set up as the hypothesis, and normal distribution is used to estimate the proportion of the products will reach this value.

Normal probability density calculates the probability density of normal distribution that data taken from a specified x value.

Normal distribution probability calculates the probability of normal distribution data falling between two specific values.

Inverse cumulative normal distribution calculates a value that represents the location within a normal distribution for a specific cumulative probability.

Student-

t probability density calculates the probability density of t distribution that data

taken from a specified x value.

Student- t distribution probability calculates the probability of t distribution data falling between two specific values.

Like t distribution, distribution probability can also be calculated for !2, F, Binomial, Poisson, and Geometric distributions.

On the initial STAT2 Mode screen, press 5 (DIST) to display the distribution menu, which contains the following items.

• 5(DIST)b(Norm) ... Normal distribution (p.44)

c(T) ... Student-t distribution (p.48)

d(!2) ... !2 distribution (p.50)

e(F) ... F distribution (p.53)

f(Binmal) ... Binomial distribution (p.57)

g(Poissn) ... Poisson distribution (p.60)

h(Geo) ... Geometric distribution (p.62)

Page 43

uCommon Distribution Functions

After drawing a graph, you can use the P-CAL function to calculate an estimated p-value for a particular x value.

The following is the general procedure for using the P-CAL function.

1. After drawing a graph, press 1 (P-CAL) to display the x value input dialog box.

2. Input the value you want for x and then press w.

• This causes the x and p values to appear at the bottom of the display, and moves the pointer to the corresponding point on the graph.

3. Pressing v or 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 i to clear the coordinate values and the pointer from the display.

# Executing an analysis function automatically

stores the x and p values in alpha variables

and P, respectively.

Page 44

k Normal Distribution

uNormal Probability Density

Normal probability density calculates the probability density of nomal distribution that data taken from a specified x value. Normal probability density is applied to standard normal distribution.

f(x) =

–

(x – µ)

(# > 0)

Perform the following key operation from the statistical data list.

5(DIST)

b(Norm)

b(P.D)

Data is specified using parameter specification. The following shows the meaning of each item.

x .................................. data

.................................. population standard deviation (

> 0)

.................................. population mean

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation or draws a graph

• Specifying # = 1 and µ = 0 specifies standard normal distribution.

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

• p ... normal probability density

# V-Window settings for graph drawing are set

automatically when the SET UP screen's [Stat Wind] setting is [Auto]. Current V-

Window settings are used for graph drawing when the [Stat Wind] setting is [Manual].

Page 45

uNormal Distribution Probability

Normal distribution probability calculates the probability of normal distribution data falling between two specific values.

p =

–

(x – µ)

a : lower boundary b : upper boundary

Perform the following key operation from the statistical data list.

5 (DIST)

b (Norm)

c (C.D)

Data is specified using parameter specification. The following shows the meaning of each item.

Lower .......................... lower boundary

Upper .......................... upper boundary

.................................. population standard deviation (# > 0)

.................................. population mean

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

# There is no graphing for normal distribution

probability.

Page 46

Calculation Result Output Example

p .................................. normal distribution probability

z:Low ........................... z:Low value (converted to standardize z score for lower

value)

z:Up ............................. z:Up value (converted to standardize z score for upper

value)

uInverse Cumulative Normal Distribution

Inverse cumulative normal distribution calculates a value that represents the location within a normal distribution for a specific cumulative probability.

f (x)dx = p

Specify the probability and use this formula to obtain the integration interval.

Perform the following key operation from the statistical data list.

5(DIST)

b(Norm)

d(Invrse)

Data is specified using parameter specification. The following shows the meaning of each item.

Tail ............................... probability value tail specification (Left, Right, Central)

Area ............................ probability value (0 < Area < 1)

.................................. population standard deviation (

> 0)

.................................. population mean

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

Tail : Left

upper boundary of integration interval

' = ?

Tail : Right

lower boundary of integration interval

' = ?

Tail : Central

upper and lower boundaries of integration interval

' = ? ( = ?

Page 47

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Examples

x ....................................... inverse cumulative normal distribution

(Tail:Left upper boundary of integration interval) (Tail:Right lower boundary of integration interval) (Tail:Central upper and lower boundaries of integration interval)

# There is no graphing for inverse cumulative

normal distribution.

Page 48

k Student-t Distribution

uStudent-t Probability Density

Student-t probability density calculates the probability density of t distribution that data taken from a specified x value.

(x) =

)

–

df+1

df + 1

Perform the following key operation from the statistical data list.

5(DIST)

c(T)

b(P.D)

Data is specified using parameter specification. The following shows the meaning of each item.

x .................................. data

df ................................. degrees of freedom (df >0)

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

p ... Student-t probability density

# Current V-Window settings are used for

graph drawing when the SET UP screen's [Stat Wind] setting is [Manual]. The VWindow settings below are set automatically

when the [Stat Wind] setting is [Auto].

Xmin = –3.2, Xmax = 3.2, Xscale = 1,

Ymin = –0.1, Ymax = 0.45, Yscale =0.1

Page 49

uStudent-t Distribution Probability

Student-t distribution probability calculates the probability of t distribution data falling between two specific values.

p =

)

df + 1

–

df+1

a : lower boundary b : upper boundary

Perform the following key operation from the statistical data list.

5(DIST)

c(T)

c(C.D)

Data is specified using parameter specification. The following shows the meaning of each item.

Lower .......................... lower boundary

Upper .......................... upper boundary

df ................................. degrees of freedom (df > 0)

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

# There is no graphing for Student-t distribution

probability.

Page 50

Calculation Result Output Example

p ... Student-t distribution probability

t:Low ... t:Low value (input lower value)

t:Up ... t:Up value (input upper value)

k !2 Distribution

u!2 Probability Density

!2 probability density calculates the probabilitty density function for the !2 distribution at a

specified x value.

(x) =

)

x e

–1

–

Perform the following key operation from the statistical data list.

5(DIST)

d(!2)

b(P.D)

Data is specified using parameter specification. The following shows the meaning of each item.

x .................................. data

df ................................. degrees of freedom (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.

Page 51

Calculation Result Output Example

p ... !2 probability density

# Current V-Window settings are used for

graph drawing when the SET UP screen's [Stat Wind] setting is [Manual]. The VWindow settings below are set automatically

when the [Stat Wind] setting is [Auto].

Xmin = 0, Xmax = 11.5, Xscale = 2, Ymin = -0.1, Ymax = 0.5, Yscale =0.1

Page 52

Distribution Probability

!2 distribution probability calculates the probability of !2 distribution data falling between two

specific values.

p =

)

x e dx

–1

–

a : lower boundary b : upper boundary

Perform the following key operation from the statistical data list.

5(DIST)

d(!2)

c(C.D)

Data is specified using parameter specification. The following shows the meaning of each item.

Lower .......................... lower boundary

Upper .......................... upper boundary

df ................................. degrees of freedom (positive integer)

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

# There is no graphing for !2 distribution

probability.

Page 53

Calculation Result Output Example

p ... !2 distribution probability

k F Distribution

u F Probability Density

probability density calculates the probability density function for the F distribution at a

specified x value.

)

–1

)

n + d

)

1 +

n + d

(x) =

–

Perform the following key operation from the statistical data list.

5(DIST)

e(F) b(P.D)

Data is specified using parameter specification. The following shows the meaning of each item.

x .................................. data

n:df .............................. numerator degrees of freedom (positive integer)

d:df .............................. denominator degrees of freedom (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.

Page 54

Calculation Result Output Example

p ... F probability density

# V-Window settings for graph drawing are set

automatically when the SET UP screen's [Stat Wind] setting is [Auto]. Current V-

Window settings are used for graph drawing when the [Stat Wind] setting is [Manual].

Page 55

u F Distribution Probability

F distribution probability calculates the probability of F distribution data falling between two

specific values.

p =

)

–1

)

n + d

)

1 +

n + d

–

a : lower boundary b : upper boundary

Perform the following key operation from the statistical data list.

5 (DIST)

e (F)

c (C.D)

Data is specified using parameter specification. The following shows the meaning of each item.

Lower .......................... lower boundary

Upper .......................... upper boundary

n:df .............................. numerator degrees of freedom (positive integer)

d:df .............................. denominator degrees of freedom (positive integer)

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

# There is no graphing for F distribution

probability.

Page 56

Calculation Result Output Example

p ... F distribution probability

Page 57

k Binomial Distribution

u Binomial Probability

Binomial probability calculates a probability at specified value for the discrete binomial distribution with the specified numtrials and probability of success on each trial.

(x) = nCxpx(1–p)

n – x

(x = 0, 1, ·······, n) p : success probability

(0 < p < 1)

n : number of trials

Perform the following key operation from the statistical data list.

5(DIST)

f(Binmal)

b(P.D)

The following shows the meaning of each item when data is specified using list specification.

Data ............................ data type

List .............................. list whose contents you want to use as specified data

Numtrial ....................... number of trials

p .................................. success probability (0 < p < 1)

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

The following shows the meaning of parameter data specification items that are different from list data specification.

x .................................. integer from 0 to n

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

# There is no graphing for binomial distribution.

Page 58

Calculation Result Output Example

p ... Binomial probability

uBinomial Cumulative Density

Binomial cumulative density calculates a cumulative probability at specified value for the discrete binomial distribution with the specified numtrials and probability of success on each trial.

Perform the following key operation from the statistical data list.

5 (DIST)

f (Binmal)

c (C.D)

The following shows the meaning of each item when data is specified using list specification.

Data ............................ data type

List .............................. list whose contents you want to use as specified data

Numtrial ....................... number of trials

p .................................. success probability (0 < p < 1)

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

The following shows the meaning of parameter data specification item that is different from list data specification.

x .................................. integer from 0 to n

Page 59

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

p ... Binomial cumulative density

Page 60

k Poisson Distribution

uPoisson Probability

Poisson probability calculates a probability at specified value for the discrete Poisson distribution with the specified mean.

f(x) =

–

(x = 0, 1, 2, ···)

: population mean (

> 0)

Perform the following key operation from the statistical data list.

5 (DIST)

g (Poissn)

b (P.D)

The following shows the meaning of each item when data is specified using list specification.

Data ............................ data type

List .............................. list whose contents you want to use as specified data

.................................. population mean (µ > 0)

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

The following shows the meaning of parameter data specification item that is different from list data specification.

x .................................. ( x

> 0)

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

p ... Poisson probability

# There is no graphing for Poisson distribution.

Page 61

u Poisson Cumulative Density

Poisson cumulative density calculates a cumulative probability at specified value for the discrete Poisson distribution with the specified mean.

Perform the following key operation from the statistical data list.

5 (DIST)

g (Poissn)

c (C.D)

The following shows the meaning of each item when data is specified using list specification.

Data ............................ data type

List .............................. list whose contents you want to use as specified data

.................................. population mean (µ > 0)

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a caluculation

The following shows the meaning of parameter data specification item that is different from list data specification.

x .................................. ( x

> 0)

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

p ... Poisson cumulative density

Page 62

k Geometric Distribution

uGeometric Probability

Geometric probability calculates a probability at specified value, the number of the trial on which the first success occurs, for the discrete geometric distribution with the specified probability of success.

(x) = p(1– p)

x – 1

(x = 1, 2, 3, ···)

Perform the following key operation from the statistical data list.

5(DIST)

h(Geo)

b(P.D)

The following shows the meaning of each item when data is specified using list specification.

Data ............................ data type

List .............................. list whose contents you want to use as specified data

p .................................. success probability (0 < p < 1)

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

The following shows the meaning of parameter data specification item that is different from list data specification.

x .................................. positive integer (x > 1)

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

p ... Geometric probability

# There is no graphing for geometric distribu-

tion.

# Positive integer number is calculated whether

list data (Data:List) or

x value (Data:variable)

is specified.

Page 63

uGeometric Cumulative Density

Geometric cumulative density calculates a cumulative probability at specified value, the number of the trial on which the first success occurs, for the discrete geometric distribution with the specified probability of success.

Perform the following key operation from the statistical data list.

5(DIST)

h(Geo)

c(C.D)

The following shows the meaning of each item when data is specified using list specification.

Data ............................ data type

List .............................. list whose contents you want to use as specified data

p .................................. success probability (0 < p < 1)

Save Res .................... list for storage of calculation results (None or List 1 to 20)

Execute ....................... executes a calculation

The following shows the meaning of parameter data specification item that is different from list data specification.

x .................................. positive integer (x > 1)

After setting all the parameters, align the cursor with [Execute] and then press the function key shown below to perform the calculation.

• 1(CALC) ... Performs the calculation.

Calculation Result Output Example

p ... Geometric cumulative density

# Positive integer number is calculated whether

list data (Data:List) or

x value (Data:variable)

is specified.

Page 64

Examples

Page 65

examples

k 1-Sample

Z Test

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as the Data parameter), be sure to input data into the list first.

3. 3(TEST)b(Z)b(1-Smpl) ... 1-Sample Z Test

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph.

Page 66

examples

Example Five new members of a football team are timed for the 100-

meter dash, yielding the following times.

A: 12.5 B: 11.6 C: 10.8 D: 12.8 E: 11.4

The average time of current team members is 11.4 seconds, with a standard deviation of 1.30. Test the null hypothesis that the times of the five new members are at the same level as current team members at the 0.05 level of significance.

Procedure

1 m STAT2

2 bc.fwbb.gwba.iw

bc.iwbb.ew

3 3(TEST) b(Z)b(1-Smpl)

4 1(LIST) c

1(!) c

bb.ew b.dw

1(LIST) bwc 1(1)c 1(None)c

5 1(CALC)

6(DRAW)

Result Screen

Since P = 0.47003508 > 0.05 (level of significance), we can't reject the null hypothesis and can conclude the times of the five new numbers is at the same level as current team members.

Page 67

examples

k 2-Sample

Z Test

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as the Data parameter) , be sure to input data into the list first.

3. 3(TEST)b(Z)c(2-Smpl) ... 2-Sample Z Test

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph.

Page 68

examples

Example A consumer group is testing camp stoves. To test the heating capacity

of a stove they measure the time required to bring 2 qt of water from 50

F to boiling (at sea level). Two competing models are under consideration. 36 stoves of each model were tested and the following results were obtained.

Model 1: mean time

o1 = 11.5min; standard deviation "1= 2.4min

Model 2: mean time

o2 = 10 min; standard deviation "2= 3 min

Is there any difference between the performances of these two models? (Use a 5% level of significance.)

Procedure

1 m STAT2

2 3(TEST) b(Z)c(2-Smpl)

3 2(VAR) c

1(!) c

c.ew dw bb.fw dgw baw dgw

1(None) c

4 1(CALC)

6(DRAW)

Result Screen

Since P = 0.01914957 < 0.05 (level of significance), we reject the null hypothesis µ1-µ2 =0 and conclude the performances of the two models are different.

Page 69

examples

k 1-Prop

Z Test

Set Up

1. On the icon menu, select STAT2.

Execution

2. 3(TEST)b(Z)d(1-Prop) ... 1-Prop Z Test

3. Set calculation parameters.

4. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph.

Page 70

examples

Example A team of eye surgeons has developed a new technique for a risky eye

operation to restore the sight of people blinded from a certain disease. Under the old method it is known that only 30% of the patients who undergo this operation recover their eyesight. Suppose that surgeons in various hospitals have performed a total of 250 operations using the new method and that 93 have been successful (the patients fully recovered their sight). Can we justify the claim that the new method is better than the old one?(Use a 1% level of significance.)

Procedure

1 m STAT2

2 3(TEST) b(Z)d(1-Prop)

3 3(>) c

.dw jdw cfaw

1(None) c

4 1(CALC)

6(DRAW)

Result Screen

Since P = 6.4915e-3 < 0.01 (level of significance), we can reject the null hypothesis that the successful probability of new operation is 0.3 and conclude that it can be said the new method obtains good results more than existing method. That is to say it is more effective by new methods.

Page 71

k 2-Prop

Z Test

Set Up

1. On the icon menu, select STAT2.

Execution

2. 3(TEST)b(Z)e(2-Prop) ... 2-Prop Z Test

3. Set calculation parameters.

4. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph.

examples

Page 72

examples

Example The County Clerk wishes to improve voter registration. One method

under consideration is to send reminders in the mail to all citizens in the county who are eligible to register. As a pilot study to determine if this method will actually improve voter registration, a random sample of 1224 potential voters was taken. Then this sample was randomly divided into two groups.

Group A : There were 612 people in this group. No reminders to

Group B : This group also contained 612 people. Reminders were

sent in the mail to each member in the group, and the number who registered to vote was 335.

The County Clerk claims that the proportion of people to register was significantly greater in group B. On the basis of this claim the clerk recommends that the project be funded for the entire population of Macek County. Use 5% level of significance to test the claim that the proportion of potential voters who registered was greater in group B, the group that received reminders.

Procedure

1 m STAT2

2 3(TEST) b(Z)e(2-Prop)

3 2(<) c

ciaw gbcw ddfw gbcw

1(None) c

4 1(CALC)

6(DRAW)

Result Screen

Since P = 8.3277e-4 < 0.05 (level of significance), we can reject the null hypothesis (P1>P2) and conclude that the County Clerk‘s claim is valid at the 0.05 level of significance.

Page 73

k 1-Sample

t Test

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as the Data parameter), be sure to input data into the list first.

3. 3(TEST)c(T)b(1-Smpl) ... 1-Sample t Test

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph.

examples

Page 74

Example A company manufactures large rocket engines used to project

satellites into space. The government buys the rockets, and the contract specifies that these engines are to use an average of 5550 lb of rocket fuel the first 15 sec operation. The company claims their engines fit specifications. To test the claim an inspector randomly selects six such engines from the warehouse. These 6 engines are fired 15 sec each and the fuel consumption for each engine is measured. For all six engines, the mean fuel consumption is

o = 5750

pounds and the standard deviation is x

n-1 = 250 lb. Is the claim

justified at the 5% level of significance?

Procedure

1 m STAT2

2 3(TEST) c(T)b(1-Smpl)

3 2(VAR) c

1(!) c

fffaw fhfaw cfaw gw

1(None) c

4 1(CALC)

6(DRAW)

Result Screen

examples

Since P = 0.107344401 > 0.05 (level of significance), we can’t reject the null hypothesis

µ=5550 and conclude that the data does not present sufficient evidence to indicate that the

average fuel consumption from the first 15 seconds of operation is different from µ =5550.

Page 75

k 2-Sample

t Test (Pooled On)

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as the Data parameter), be sure to input data into the list first.

3. 3(TEST)c(T)c(2-Smpl) ... 2-Sample t Test

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph.

examples

Page 76

Example Two different processes (Type A and Type B) are used to produce

tinplate. The following values show the weights of samples produced by each process.

Type A: 105 108 86 103 103 107 124 124 Type B: 89 92 84 97 103 107 111 97

Using the level of significance 0.05, test the null hypothesis that the tinplate produced by the two processes are of the same level.

Procedure

1 m STAT2

2 bafwbaiwigwbadw

badwbahwbcewbcew

ijwjcwiewjhw badwbahwbbbwjhw

3 3(TEST) c(T)c(2-Smpl)

4 1(LIST) c

1(!) c 1(LIST) bwc 1(LIST) cwc 1(1)c 1(1)c 1(On) c 1(None) c

5 1(CALC)

6(DRAW)

Result Screen

examples

Since P = 0.08602732 > 0.05 (level of significance), we can not reject the null hypothesis and conclude that the tinplate produced by the two processes are of the same level.

Page 77

k 2-Sample

t Test (Pooled Off)

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as the Data parameter) , be sure to input data into the list first.

3. 3(TEST)c(T)c(2-Smpl) ... 2-Sample t Test

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph.

examples

Page 78

examples

Example Two competing headache remedies claim to give fast-acting relief. An

experiment was performed to compare the mean lengths of time required for bodily absorption of brand A and brand B headache remedies. 12 people were randomly selected and given an oral dosage of brand A. Another 12 were randomly selected and given an equal dosage of brand B. The length of time in minutes for the drugs to reach a specified level in the blood was recorded. The means, standard deviations, and sizes of the two samples follow;

Brand A:

o1 = 19.8; x1

n-1 = 8.6; n1 = 12

Brand B:

o2 = 19 ; x2

n-1 = 6 ; n2 = 12

Past experience with the drug composition of the two remedies permits researchers to assume the standard deviations of the two time distributions are approximately equal. Let us use a 5% level of significance to test the null hypothesis that there is no difference in the mean time required for bodily absortion.

Procedure

1 m STAT2

2 3(TEST) c(T)c(2-Smpl)

3 2(VAR) c

1(!) c

bj.iw i.gw bcw bjw gw bcw

2(Off) c 1(None) c

4 1(CALC)

6(DRAW)

Result Screen

Since P = 0.79431564 > 0.05 (level of significance), the difference is in acceptance region. We conclude that the data does not contain sufficient evidence to reject H0. Therefore, at the 5% level of significance we conclude that there is no difference in the mean times.

Page 79

examples

k LinearReg

t Test

Set Up

1. On the icon menu, select STAT2.

Execution

2. Input data into the list.

3. 3(TEST)c(T)d(LinReg) ... LinearReg t Test

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

Page 80

examples

Example A survey conducted by a city on its real estate investments revealed

the following data on the relationship between area and price. Does the data indicate that the value

1, that is, the slope of the popula-

tion regression line, is not zero, which would mean that x (Area) can be use as a predictor of y (Sales Price)? Use a 5% level of significance.

Procedure

1 m STAT2

2jwbfwbawbbwbaw

dgwiaweewffwdfw

3 3(TEST) c(T)d(LinReg)

4 1(!) c

1(LIST) bwc 1(LIST) cwc 1(1)c 1(None) c

5 1(CALC)

Result Screen

Since P = 6.4531e-3 < 0.05 (level of significance), we can reject the null hypothesis

1 ! 0

and conclude that there is a linear relationship y =

0 + #1x + $ between area(x) and price(y).

x Area (square feet) 9 15 10 11 10

y Sales Price (% $1000) 36 80 44 55 35

Page 81

examples

Test

Set Up

1. On the icon menu, select STAT2.

Execution

2. 3(TEST)d(

)...

Test

3. Input data into the Matrix.

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph.

Page 82

examples

Example A certain club collected data on the attendance at meetings by married

status, and obtained the data shown below.

Married Divorced Widowed Single Row Totals

Often Absent 34 16 14 36 100

Seldom Absent 64 34 20 82 200

Never Absent 52 50 16 82 200

Column Totals 150 100 50 200 500

Test the

null hypothesis that two phenomena are independent using

the level of significance of 0.05.

Procedure

1 m STAT2

2 3(TEST) d( &2)

3 2('MAT)

1(DIM)

dw eww

dewbgwbewdgw gewdewcawicw fcwfawbgwicw

4 1(MAT) v(A) wc

1(MAT) l(B) wc 1(None) c

5 1(CALC)

6(DRAW)

Result Screen

Since P = 0.17546141 > 0.05 (level of significance), we can not reject the null hypothesis and conclude that there the two phenomena are independent of each other.

Page 83

k 2-Sample

F Test

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as the Data parameter), be sure to input data into the list first.

3. 3(TEST)e(F)... 2-Sample F Test

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph.

examples

Page 84

Example There are two possible routes to get to the airport from a certain

company, and so manager wants to determine which route is the fastest in order to make a plane that leaves at 7 o'clock. One route was researched five times and then the other route was researched five times, producing the data shown below.

mean of x1 = 33.8min mean of x2 = 34.2min x1

n-1 = 16.4min x2"n-1 = 6.1min

Test the null hypothesis that the two routes have the same travel time variance using the level of significance of 0.10.

Procedure

1 m STAT2

2 3(TEST) e( F)

3 2(VAR) c

1(!) c

bg.ew fw g.bw fw

1(None) c

4 1(CALC)

6(DRAW)

Result Screen

examples

Since P = 0.08144232 < 0.1 (level of significance), we can reject the null hypothesis and conclude that two routes don’t have same travel time variance.

Page 85

examples

k One-Way ANOVA

Set Up

1. On the icon menu, select STAT2.

Execution

2. Input data into the list.

3. 3(TEST)f(ANOVA) ... Analysis of Variance (ANOVA)

4. Set calculation parameters. Specify 1 for How Many setting.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

Page 86

examples

Example A psychologist is studying pattern recognition skills under four

laboratory settings. In each setting, a fourth-grade child is given a pattern recognition test with ten patterns to identify. In setting A, the child is given praise for each correct answer and no comment about wrong answers. In setting B, the child is given criticism for each wrong answer and no comment about correct answers. In setting C, the child is given no praise or criticism but the observer expresses interest in what the child is doing. In setting D, the observer remains silent in an adjacent room watching the child through a one-way mirror. A random sample of fourth-grade children was used, and each child participated in the test only once. The test scores (number correct) for each group follow. Find F

0.01, the critical values for an ' =

0.01 level of significance. Does the test indicate we should accept or reject the null hypothesis?

Group A = {10, 8, 7, 9, 11} Group B = { 2, 5, 3, 3, 4} Group C = { 9, 3, 7, 8, 5, 6} Group D = { 5, 7, 3, 6, 7}

Procedure

1 m STAT2

2bwbwbwbwbwcwcwcwcwcw

dwdwdwdwdwdwewewewewew

bawiwhwjwbbwcwfwdwdwew jwdwhwiwfwgwfwhwdwgwhw

3 3(TEST) f(ANOVA)

4 1(1)c

1(LIST) bwc 1(LIST) cwc 1(None) c

5 1(CALC)

Result Screen

Since P = 7.992708666e-4 < 0.01 (level of significance), we can reject the null hypothesis and conclude that the laboratory setting affects the mean score.

Page 87

examples

k Two-Way ANOVA

Set Up

1. On the icon menu, select STAT2.

Execution

2. Input data into the list.

3. 3(TEST)f(ANOVA) ... Analysis of Variance (ANOVA)

4. Set calculation parameters. Specify 2 for How Many setting.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

6(DRAW) ... Draws graph

Example The nearby table shows measurement results for a metal product

produced by a heat treatment process based on two treatment levels: time (A) and temperature (B). The experiments were repeated twice each under identical conditions.

Perform analysis of variance on the following null hypothesis, using a significance level of 5%.

H0 : No change in strength due to time H0 : No change in strength due to heat treatment temperature H0 : No change in strength due to interaction of time and heat treatment temperature

B (Heat Treatment Temperature) B1 B2

A1 113 , 116

133 , 131

139 , 132

126 , 122

A (Time)

Page 88

Procedure

1 m STAT2

2bwbwbwbwcwcwcwcwe

bwbwcwcwbwbwcwcwe bbdwbbgwbdjwbdcwbddwbd bwbcgwbccw

3 3(TEST) f(ANOVA)

4 2(2)c

1(LIST) bwc 1(LIST) cwc 1(LIST) dwc 1(None) c

5 1(CALC)

6(DRAW)

Result Screen

• Time differential (A) :

Since P=0.2458019517 > 0.05 (level of significance), we can not reject the null hypothesis.

• Temperature differential (B) :

Since P=0.04222398836 < 0.05 (level of significance), we can reject the null hypothesis.

• Interaction (A % B) :

Since P=2.78169946e-3 < 0.05 (level of significance), we can reject the null hypothesis.

The above test indicates that the time differential is not significant, the temperature differential is significant, and interaction is highly significant.

examples

Page 89

examples

k 1-Sample Z Interval

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as the Data parameter), be sure to input data into the list first.

3. 4(INTR)b(Z)b(1-Smpl)... 1-Sample Z Interval

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

Page 90

examples

Example In a random sample of size n = 20 from a normal population

with the standard deviation ! = 15 and the mean

o = 64.3,

construct a 95% confidence interval for the population mean µ.

Procedure

1 m STAT2

2 4(INTR) b(Z)b(1-Smpl)

3 2(VAR) c

.jfw bfw ge.dw caw

1(None)c

4 1(CALC)

Result Screen

Thus, the 95% confidence interval for µ becomes 57.7<µ<70.9 .

Page 91

examples

k 2-Sample Z Interval

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as Data parameter), be sure to input data into the list first.

3. 4(INTR)b(Z)c(2-Smpl)... 2-Sample Z Interval

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

Page 92

examples

Example Construct a 94% confidence interval for the difference between the

mean lifetimes of two kinds of light bulbs(

µ1-µ

2), given that random

sample of 40 light bulbs of the first kind lasted on the average for 418 hours of continuous use and 50 light bulbs of the second kind lasted on the average for 402 hours of continuous use. The population standard deviations are known to be

1 = 26 and !2 = 22.

Procedure

1 m STAT2

2 4(INTR) b(Z)c(2-Smpl)

3 2(VAR) c

.jew cgw ccw ebiw eaw eacw faw

1(None)c

4 1(CALC)

Result Screen

Thus, the 94% confidence interval for

µ1-µ2

becomes 6.3<

µ1-µ

2<25.7.

Page 93

examples

k 1-Prop Z Interval

Set Up

1. On the icon menu, select STAT2.

Execution

2. 4(INTR)b(Z)d(1-Prop)... 1-Prop Z Interval

3. Set calculation parameters.

4. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

Page 94

examples

Example Suppose 800 students were selected at random from a student body

of 20,000 and given shots to prevent a certain type of flu. All 800 students were exposed to the flu and 600 of them did not get the flu. Let p present the probability that the shot will be successful for any single student selected at random from the entire population of 20,000. Let q be the probability that the shot is not successful.

a) What are the point estimates for p and q? b) Find a 99% confidence interval for p.

Procedure

1 m STAT2

2 4(INTR) b(Z)d(1-Prop)

3 .jjw

gaaw iaaw

1(None)c

4 1(CALC)

Result Screen

a) The point of estimates for p = 0.75, and the point of

estimates for q = 1-p = 0.25.

b) Thus, the 99% confidence interval for p becomes 0.71<p<0.79 .

Page 95

examples

k 2-Prop Z Interval

Set Up

1. On the icon menu, select STAT2.

Execution

2. 4(INTR)b(Z)e(2-Prop)... 2-Prop Z Interval

3. Set calculation parameters.

4. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

Page 96

examples

Example If 132 of 200 male voters and 90 of 150 female voters favor a certain

candidate running for governor of Texas, find a 99% confidence interval for the difference between the actual proportions of male and female voters who favor the candidate.

Procedure

1 m STAT2

2 4(INTR) b(Z)e(2-Prop)

3 .jjw

bdcw caaw jaw bfaw

1(None)c

4 1(CALC)

Result Screen

Let p1 the actual propotion of male voters who favor the candidate. Let p2 the actual propotion of female voters who favor the candidate. Thus, the 99% confidence interval for (p1-p2) is -0.074<p1-p2<0.194.

Page 97

examples

k 1-Sample t Interval

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as Data parameter), be sure to input data into the list first.

3. 4(INTR)c(T)b(1-Smpl)... 1-Sample t Interval

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

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examples

Example A paint manufacture wants to determine the average drying time of a

new interior wall paint. If for 12 test areas of equal size he obtained a mean drying time of 66.3 minutes and a standard deviation of 8.4 minutes, construct a 95% confidence interval for the population mean

Procedure

1 m STAT2

2 4(INTR) c(T)b(1-Smpl)

3 2(VAR) c

.jfw gg.dw i.ew bcw

1(None)c

4 1(CALC)

Result Screen

Thus, the 95% confidence interval for µ becomes 61.0<µ<71.6 .

Page 99

examples

k 2-Sample t Interval (Pooled On)

Set Up

1. On the icon menu, select STAT2.

Execution

2. When using list data (List is selected as the Data parameter), be sure to input data into the list first.

3. 4(INTR)c(T)c(2-Smpl) ... 2-Sample t Interval

4. Set calculation parameters.

5. Align the cursor with [Execute]

1(CALC) ... Performs calculation.

Page 100

examples

Example Factory A and Factory B both manufacture the same item. To compare

the productivity of the two factories, the production data shown below was collected for 10 items at Factory A and nine items at Factory B.

Factory A: 78, 80, 79, 83, 82, 85, 78, 74, 76, 84 (kg/h) Factory B: 81, 84, 82, 88, 86, 83, 78, 84, 89 (kg/h)

Construct a 95% confidence interval for the difference between

1 and

2 when the following is assumed: ! 1

, Factory A population

distribution is N(

), and Factory B population distribution is N(

Procedure

1 m STAT2

2 hiwiawhjwidwicw

ifwhiwhewhgwiew

ibwiewicwiiwigw

idwhiwiewijw

3 4(INTR) c(T)c(2-Smpl)

4 1(LIST) c

.jfw

1(LIST)bwc 1(LIST)cwc 1(1)c 1(1)c 1(On)c 1(None)c

5 1(CALC)

Result Screen

Thus, the 95% confidence interval for

µ1-µ2

becomes -7.4<

µ1-µ

2<-0.6.