Casio ALGEBRA FX 2 0 PLUS User Manual

ALGEBRA FX 2.0 PLUS

FX 1.0 PLUS

User’s Guide

(Additional Functions)

http://world.casio.com/edu_e/

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ALGEBRA FX 2.0 PLUS

FX 1.0 PLUS

(Additional Functions)

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20010101

Contents

Chapter 1 Advanced Statistics Application

1-1 Advanced Statistics (STAT) .............................................................. 1-1-1

1-2 Tests (TEST) .................................................................................... 1-2-1

1-3 Confidence Interval (INTR)............................................................... 1-3-1

1-4 Distribution (DIST)............................................................................ 1-4-1

Chapter 2 Financial Calculation (TVM)

2-1 Before Performing Financial Calculations ........................................ 2-1-1

2-2 Simple Interest ................................................................................. 2-2-1

2-3 Compound Interest........................................................................... 2-3-1

2-4 Cash Flow (Investment Appraisal).................................................... 2-4-1

2-5 Amortization ..................................................................................... 2-5-1

2-6 Interest Rate Conversion.................................................................. 2-6-1

2-7 Cost, Selling Price, Margin ............................................................... 2-7-1

2-8 Day/Date Calculations...................................................................... 2-8-1

2-9 Depreciation ..................................................................................... 2-9-1

2-10 Bonds ............................................................................................. 2-10-1

2-11 TVM Graph..................................................................................... 2-11-1

Chapter 3 Differential Equations

3-1 Using the DIFF EQ Mode ................................................................. 3-1-1

3-2 Differential Equations of the First Order ........................................... 3-2-1

3-3 Linear Differential Equations of the Second Order ........................... 3-3-1

3-4 Differential Equations of the Nth Order ............................................ 3-4-1

3-5 System of First Order Differential Equations .................................... 3-5-1

Chapter 4 E-CON

4-1 E-CON Overview.............................................................................. 4-1-1

4-2 EA-100 Setup ................................................................................... 4-2-1

4-3 Setup Memory .................................................................................. 4-3-1

4-4 Program Converter........................................................................... 4-4-1

4-5 Starting a Sampling Operation ......................................................... 4-5-1

Index

20010101

Chapter

Advanced Statistics Application

1-1 Advanced Statistics (STAT)

1-2 Tests (TEST)

1-3 Confidence Interval (INTR)

1-4 Distribution (DIST)

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1-1-1

Advanced Statistics (STAT)

1-1 Advanced Statistics (STAT)

uFunction Menu

The following shows the function menus for the STAT Mode list input screen.

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 (page1-2-1)

• 4(INTR) ... Confidence interval (page1-3-1)

• 5(DIST) ... Distribution (page1-4-1)

SORT and JUMP functions are located in the TOOL menu (6(g)1(TOOL)).

uCalculation of the Coefficient of Determination (r²) and MSE

You can use the STAT Mode to calculate the coefficient of determination (r²) for quadratic

regression, cubic regression, and quartic regression. The following types of MSE

calculations are also available for each type of regression.

(yi – (axi+ b))²

• Linear Regression ...

• Quadratic Regression ...

• Cubic Regression ...

• Quartic Regression ...

MSE =

n – 2

i=1

(yi – (ax ²

+ bxi+ c))²

+ bx ²+ cxi +d ))²

(yi – (ax⁴+ bx ³+ cx ²+ dxi + e))²

n – 3

i=1

(yi – (ax³

n – 4

i=1

n – 5

i=1

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1-1-2

Advanced Statistics (STAT)

(yi – (a + b ln xi ))²

• Logarithmic Regression ...

MSE =

n – 2

i=1

(ln yi – (ln a + bxi ))²

(ln yi – (ln a + b ln xi ))²

(yi – (a sin (bxi + c) + d ))²

• Exponential Repression ...

• Power Regression ...

• Sin Regression ...

n – 2

i=1

n – 2

i=1

n – 2

i=1

• Logistic Regression ...

–

1 + ae^-bx

n – 2

i=1

uEstimated Value Calculation for Regression Graphs

The STAT Mode also includes aY-CAL function that uses regression to calculate the

estimated y-value for a particular x-value after graphing a paired-variable statistical

regression.

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

1. After drawing a regression graph, press 6(g)2(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.

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1-1-3

Advanced Statistics (STAT)

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.

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 normal regression formula copy function that lets you copy the regression

calculation result screen after drawing a statistical graph (such as Scatter Plot), the STAT Mode

also has a function that lets you copy the regression formula obtained as the result of a regression

calculation. To copy a resulting regression formula, press 6(COPY).

k Tests, Confidence Interval, and Distribution Calculations

The STAT Mode includes functions for performing tests, and confidence interval and distribution

calculations.You can find explanations of each of these functions in the following sections: 1-2

Tests, 1-3 Confidence Interval, and 1-4 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

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

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Advanced Statistics (STAT)

uCommon Functions

• The symbol “ ” appears in the upper right corner of the screen while execution of a

■

calculation 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

parameter setting screen. Pressing ! i(QUIT) returns to the top of list input screen.

· 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 disabled, except for the graph 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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1-2-1

Tests (TEST)

1-2 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 the 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 proportion 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 the population standard deviations

are unknown.

LinearReg t Test calculates the strength of the linear association of paired data.

χ 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 are two independent variables and one dependent

variable.

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1-2-2

Tests (TEST)

The following pages explain various statistical calculation methods based on the principles

described above. Details concerning statistical principles and terminology can be found in

any standard statistics textbook.

On the initial STAT Mode screen, press 3(TEST) to display the test menu, which contains

the following items.

• 3(TEST)b(Z) ... Z Tests (p. 1-2-2)

c(T) ... t Tests (p. 1-2-10)

d(χ²) ... χ²Test (p. 1-2-18)

e(F) ... 2-Sample F Test (p. 1-2-20)

f(ANOVA) ... ANOVA (p. 1-2-22)

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 ZTest

This test is used when the population standard deviation is known to test the hypothesis. The

1-Sample Z Test is applied to the normal distribution.

o – µ₀

o : mean of sample

: assumed population mean

σ : population standard deviation

Z =

µo

n : size of sample

# The following V-Window settings are used for

drawing the graph.

# Executing an analysis function automatically

stores the z and p values in alpha variables Z

and P, respectively.

Xmin = –3.2, Xmax = 3.2, Xscale = 1,

Ymin = –0.1, Ymax = 0.45, Yscale = 0.1

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1-2-3

Tests (TEST)

Perform the following key operations from the statistical data list.

3(TEST)

b(Z)

b(1-Smpl)

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

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

µ .................................. population mean value test conditions (“G µ0” specifies

two-tail test, “< µ0” specifies lower one-tail test, “> µ0

”

specifies upper one-tail test.)

µ0

................................. assumed population mean

σ .................................. population standard deviation (σ > 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.

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

• 6(DRAW) ... Draws the graph.

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1-2-4

Tests (TEST)

Calculation Result Output Example

µG11.4 ........................ direction of test

z .................................. z score

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

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

xσn-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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1-2-5

Tests (TEST)

u2-Sample Z Test

This test is used when the standard deviations for two populations are known to test the

hypothesis. The 2-Sample Z Test is applied to the normal distribution.

: mean of sample 1

: mean of sample 2

: population standard deviation of sample 1

: population standard deviation of sample 2

: size of sample 1

o₁– o₂

Z =

σ₁²σ₂²

n₁n₂

: size of sample 2

Perform the following key operations 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

µ1

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

tail test, “< µ2” specifies one-tail test where sample 1 is

smaller than sample 2, “> µ2” specifies one-tail test where

sample 1 is greater than sample 2.)

................................. population standard deviation of sample 1 (σ1 > 0)

................................. population standard deviation of sample 2 (σ2 > 0)

List(1) .......................... list whose contents you want to use as sample 1 data

(List 1 to 20)

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

(List 1 to 20)

Freq(1) ........................ frequency of sample 1 (1 or List 1 to 20)

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

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1-2-6

Tests (TEST)

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

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

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

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

¹G

µ2

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

z ...................................z score

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

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

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

1σn-1 ............................ standard deviation of sample 1

(Displayed only for Data: List setting.)

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

(Displayed only for Data: List setting.)

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

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

# [Save Res] does not save the µ1 condition in

line 2.

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1-2-7

Tests (TEST)

u1-Prop Z Test

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

applied to the normal distribution.

: expected sample proportion

– p₀

n : size of sample

Z =

p₀(1– p₀)

Perform the following key operations 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.)

................................. 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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1-2-8

Tests (TEST)

u2-Prop Z Test

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

normal distribution.

x₁x₂

–

: data value of sample 1

: data value of sample 2

: size of sample 1

n₁n₂

Z =

p(1– p )

: size of sample 2

n₁n₂

ˆp : estimated sample proportion

Perform the following key operation from the statistical data list.

3(TEST)

b(Z)

e(2-Prop)

................................. sample proportion test conditions (“G p2” specifies two-tail

test, “< p2” specifies one-tail test where sample 1 is smaller

than sample 2, “> p2” specifies one-tail test where sample 1

is greater than sample 2.)

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

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

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

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

2020010111001101

1-2-9

Tests (TEST)

1>p2 ............................ direction of test

z .................................. z score

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

pˆ1 ................................. estimated proportion of sample 1

pˆ2 ................................. estimated proportion of sample 2

pˆ .................................. estimated sample proportion

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

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

# [Save Res] does not save the p1 condition in

line 2.

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1-2-10

Tests (TEST)

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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1-2-11

Tests (TEST)

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 tTest is applied to t-distribution.

o – µ₀

: mean of sample

: assumed population mean

t =

σ^n–1

xσn-1 : sample standard deviation

: size of sample

Perform the following key operations from the statistical data list.

3(TEST)

c(T)

b(1-Smpl)

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

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

µ .................................. population mean value test conditions (“G µ0” specifies two-

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

upper one-tail test.)

µ0

................................. assumed population mean

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

xσn-1 ............................. sample standard deviation (xσn-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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Tests (TEST)

Calculation Result Output Example

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

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

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

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

xσn-1 ............................. sample standard deviation

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

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

line 2.

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1-2-13

Tests (TEST)

u2-Sample t Test

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

are unknown. The 2-Sample t Test is applied to t-distribution.

The following applies when pooling is in effect.

:mean of sample 1

:mean of sample 2

o₁– o₂

1σn-1 :standard deviation of

sample 1

2σn-1 :standard deviation of

sample 2

x_pσ_n–1

n₁n₂

(n₁–1)x₁σ _n–1+(n₂–1)x₂σ_n–1

x_pσ_n–1

n₁+ n₂– 2

:size of sample 1

:size of sample 2

df = n1 + n2 – 2

pσn-1 :pooled sample standard

deviation

df :degrees of freedom

The following applies when pooling is not in effect.

:mean of sample 1

:mean of sample 2

o₁– o₂

t =

x₁σ_n–1x₂σ_n–1

1σn-1 :standard deviation of

sample 1

2σn-1 :standard deviation of

sample 2

n₁

n₂

df =

C²(1–C )²

:size of sample 1

:size of sample 2

n₁–1 n₂–1

df :degrees of freedom

x₁σ_n–1

n₁

C =

x₁σ_n–1x₂σ_n–1

n₁

n₂

Perform the following key operations from the statistical data list.

3(TEST)

c(T)

c(2-Smpl)

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1-2-14

Tests (TEST)

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

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

µ1

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

test, “< µ2” specifies one-tail test where sample 1 is smaller

than sample 2, “> µ2” 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 1 to 20)

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

(List 1 to 20)

Freq(1) ........................ frequency of sample 1 (1 or List 1 to 20)

Freq(2) ........................ frequency of sample 2 (1 or List 1 to 20)

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.

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

1σn-1 ............................ standard deviation (x1σn-1 > 0) of sample 1

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

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

2σn-1 ............................ standard deviation (x2σn-1 > 0) of sample 2

................................. 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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Tests (TEST)

Calculation Result Output Example

µ1Gµ2 ........................... direction of test

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

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

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

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

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

1σn-1 ............................ standard deviation of sample 1

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

pσn-1 ............................ pooled sample standard deviation (Displayed only when Pooled:

On setting.)

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

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

# [Save Res] does not save the µ1 condition in

line 2.

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Tests (TEST)

uLinearReg t Test

LinearReg t Test treats paired-variable data sets as (x, y) pairs, and uses the method of

least squares to determine the most appropriate a, b coefficients of the data for the

regression formula y = a + bx. It also determines the correlation coefficient and t value, and

calculates the extent of the relationship between x and y.

: intercept

(x – o)(y – p)

: slope of the line

: size of sample (n > 3)

: correlation coefficient

n – 2

i=1

b =

a = p – bo

t = r

ⁿ(x – o)²

1 – r²

i=1

r²: coefficient of

determination

Perform the following key operations 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.

β & ρ ............................ 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 (List 1 to 20)

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

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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Tests (TEST)

Calculation Result Output Example

β G 0 & ρ 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

r²................................. 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.

# 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.

# [Save Res] does not save the β & ρ

conditions in line 2.

20010101

1-2-18

Tests (TEST)

k χ Test

χ²Test sets up a number of independent groups and tests hypothesis related to the

proportion of the sample included in each group. The χ²Test is applied to dichotomous

variables (variable with two possible values, such as yes/no).

Expected counts

x_ij

Σ Σ

i=1

j=1

F_ij=

x_ij

ΣΣ

i=1 j=1

(xij – Fij)²

χ²=

ΣΣ

i=1 j=1

Perform the following key operations from the statistical data list.

3(TEST)

d(χ²)

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.

200101101

1-2-19

Tests (TEST)

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

χ ................................. χ value

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

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

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

• 1(CHI) ... Displays χ value.

Pressing 1 (CHI) displays the χ 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 χ²value.

• 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.

# Executing an analysis function automatically

stores the χ²and p values in alpha variables

C and P, respectively.

# 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

20010101

1-2-20

Tests (TEST)

k 2-Sample F Test

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

to the F distribution.

x₁σ_n–1

F =

x₂σ_n–1

Perform the following key operations 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

................................. population standard deviation test conditions (“G σ2”

specifies two-tail test, “< σ2” specifies one-tail test where

sample 1 is smaller than sample 2, “> σ2” 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 1 to 20)

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

(List 1 to 20)

Freq(1) ........................ frequency of sample 1 (1 or List 1 to 20)

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

1σn-1 ............................ standard deviation (x1σn-1 > 0) of sample 1

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

2σn-1 ............................ standard deviation (x2σn-1 > 0) of sample 2

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

200101101

1-2-21

Tests (TEST)

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

1Gσ2 .......................... direction of test

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

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

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

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

1σn-1 ............................ standard deviation of sample 1

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

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

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

# Executing an analysis function automatically

stores the F and p values in alpha variables

F and P, respectively.

# [Save Res] does not save the σ1 condition in

line 2.

# V-Window settings are automatically

optimized for drawing the graph.

20010101

1-2-22

Tests (TEST)

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 are two independent variables and one dependent

variable.

Perform the following key operations 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

levels)

Factor A....................... category list (List 1 to 20)

Dependnt .................... list to be used for sample data (List 1 to 20)

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

16)*¹

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 (List 1 to 20)

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.

*¹[Save Res] saves each vertical column of the

table into its own list. The leftmost column is

saved in the specified list, and each

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.

subsequent column to the right is saved in

200101101

1-2-23

Tests (TEST)

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 × Factor B df value, SS value, MS value, F value,

p-value

*Line 3 does not appear when there is only one observation

in each cell.

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.TheY-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.

# Using the TRACE function automatically

stores the number of conditions to alpha

variable A and the mean value to variable M,

respectively.

# Graphing is available with Two-Way ANOVA

only. V-Window settings are performed

automatically, regardless of SET UP screen

settings.

20010101

1-2-24

Tests (TEST)

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.

B (Heat Treatment Temperature)

A (Time)

113

133

116 139

131 126

132

122

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 is not rejected.

• 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 is rejected.

• 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 is rejected.

The above test indicates that the time differential is not significant, the temperature

differential is significant, and interaction is highly significant.

200101101

1-2-25

Tests (TEST)

uInput Example

uResults

20010101

1-3-1

Confidence Interval (INTR)

1-3 Confidence Interval (INTR)

A confidence interval is a range (interval) that includes a statistical value, usually the

population mean.

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 difficult 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 the population standard deviation is known.

2-Sample Z Interval calculates the confidence interval for the difference between two

population means when the population standard deviations of two samples are known.

1-Prop Z Interval calculates the confidence interval for an unknown proportion of

successes.

2-Prop Z Interval calculates 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. 1-3-3)

c(T) ... t intervals (p. 1-3-8)

# There is no graphing for confidence interval

functions.

200101101

1-3-2

Confidence Interval (INTR)

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).

20010101

1-3-3

Confidence Interval (INTR)

k Z Interval

u1-Sample Z Interval

1-Sample Z Interval calculates the confidence interval for an unknown population mean

when the population standard deviation is known.

The following is the confidence interval.

Left = o – Z

Right = o + Z

However, α is the level of significance. The value 100 (1 – α) % is the confidence level.

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

Perform the following key operations 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)

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

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

(List 1 to 20)

Freq ............................. sample 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

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)

200101101

1-3-4

Confidence Interval (INTR)

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

xσn-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 population standard deviations of two samples are known.

The following is the confidence interval. The value 100 (1 – α) % is the confidence level.

σ₁²σ₂²

: mean of sample 1

: mean of sample 2

: population standard deviation

of sample 1

: population standard deviation

of sample 2

Left = (o₁– o₂) – Z

n₁n₂

σ₁²σ₂²

Right = (o₁– o₂) + Z

n₁n₂

: size of sample 1

: size of sample 2

Perform the following key operations from the statistical data list.

4(INTR)

b(Z)

c(2-Smpl)

200101101

1-3-5

Confidence Interval (INTR)

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 of sample 1 (σ1 > 0)

................................. population standard deviation of sample 2 (σ2 > 0)

List(1) .......................... list whose contents you want to use as data of sample 1

(List 1 to 20)

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

(List 1 to 20)

Freq(1) ........................ frequency of sample 1 (1 or List 1 to 20)

Freq(2) ........................ frequency of sample 2 (1 or List 1 to 20)

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.

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

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

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

................................. 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)

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

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

1σn-1 ............................ standard deviation of sample 1

(Displayed only for Data: List setting.)

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

(Displayed only for Data: List setting.)

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

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

200101101

1-3-6

Confidence Interval (INTR)

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 – α) % is the confidence level.

n : size of sample

x : data

1 x

Left = – Z

1–

n n

1 x

1–

Right = + Z

n n

Perform the following key operations 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

200101101

1-3-7

Confidence Interval (INTR)

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 – α) % is the confidence level.

1, n2 : size of sample

1, x2 : data

x₁

x₂

n₂

¹1–

²1–

n₁

n₁n₂

x₁

n₁

x₂

n₂

Left =

–

– Z

n₁

n₂

x₁

x₂

n₂

¹1–

²1–

n₁

n₁n₂

x₁

n₁

x₂

n₂

Right =

–

n₁

n₂

Perform the following key operations 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)

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

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

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

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

200101101

1-3-8

Confidence Interval (INTR)

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

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

ˆp1 ................................. estimated sample propotion for sample 1

ˆp2 ................................. estimated sample propotion for sample 2

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

................................. 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 – α) % is the confidence level.

σ_n–1

Left = o– t_{n – 1}

σ_n–1

Right = o+ t_{n – 1}

Perform the following key operations 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

(List 1 to 20)

Freq ............................. sample 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

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

from list data specification.

200101101

1-3-9

Confidence Interval (INTR)

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

xσn-1 ............................. sample standard deviation (xσn-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

xσn-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 – α) % is

the confidence level.

Left = (o – o )– t_{n +n}

x_pσ_n–1

₂–2

n₁n₂

Right = (o – o )+ t_{n +n}

x_pσ_n–1

₂–2

n₁n₂

(n₁–1)x₁σ _n–1+(n₂–1)x₂σ_n–1

x_pσ_n–1

n₁+ n₂– 2

200101101

1-3-10

Confidence Interval (INTR)

The following confidence interval applies when pooling is not in effect. The value

100 (1 – α) % is the confidence level.

x₁σ_n–1x₂σ_n–1

Left = (o₁– o₂)– t_df

n₁

n₂

x₁σ_n–1x₂σ_n–1

Right = (o₁– o₂)+ t_df

n₁

n₂

df =

(1–C)²

n₂–1

C²

n₁–1

x₁σ_n–1

n₁

C =

x₁σ_n–1x₂σ_n–1

n₁

n₂

Perform the following key operations 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 1 to 20)

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

(List 1 to 20)

Freq(1) ........................ frequency of sample 1 (1 or List 1 to 20)

Freq(2) ........................ frequency of sample 2 (1 or List 1 to 20)

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.

200101101

1-3-11

Confidence Interval (INTR)

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

1σn-1 ............................ standard deviation (x1σn-1 > 0) of sample 1

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

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

2σn-1 ............................ standard deviation (x2σn-1 > 0) of sample 2

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

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

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

1σn-1 ............................ standard deviation of sample 1

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

pσn-1 ............................ pooled sample standard deviation

(Displayed only when Pooled: On setting.)

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

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

20010101

1-4-1

Distribution (DIST)

1-4 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 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 t probability density 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 χ , F, Binomial, Poisson,

and Geometric distributions.

On the initial STAT Mode screen, press 5 (DIST) to display the distribution menu, which

contains the following items.

• 5(DIST)b(Norm) ... Normal distribution (p. 1-4-3)

c(T) ... Student-t distribution (p. 1-4-7)

d(χ ) ... χ distribution (p. 1-4-9)

e(F) ... F distribution (p. 1-4-12)

f(Binmal) ... Binomial distribution (p. 1-4-16)

g(Poissn) ... Poisson distribution (p. 1-4-19)

h(Geo) ... Geometric distribution (p. 1-4-21)

20010101

1-4-2

Distribution (DIST)

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 X

and P, respectively.

20010101

1-4-3

Distribution (DIST)

k Normal Distribution

uNormal Probability Density

Normal probability density calculates the probability density of nomal distribution from a

specified x value. Normal probability density is applied to standard normal distribution.

µ)²

(σ > 0)

σ ²

f(x) =

e^–^{(x –}

2πσ

Perform the following key operations 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

σ .................................. standard deviation (σ > 0)

µ .................................. mean

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

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

• Specifying σ = 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].

200101101

1-4-4

Distribution (DIST)

uNormal Distribution Probability

Normal distribution probability calculates the probability of normal distribution data falling

between two specific values.

µ)²

a : lower boundary

b : upper boundary

^be^–^{(x –}dx

σ ²

p =

∫

2πσ

Perform the following key operations 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

σ .................................. standard deviation (σ > 0)

µ .................................. 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.

200101101

1-4-5

Distribution (DIST)

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

f (x)dx = p

∫

−∞

∫

Tail: Left

upper

boundary of

integration

interval

Tail: Right

lower

boundary of

integration

interval

Tail: Central

upper and

lower

boundaries

of integration

interval

α = ?

α = ? β = ?

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

Perform the following key operations 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)

σ .................................. standard deviation (σ > 0)

µ .................................. mean

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

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

200101101

1-4-6

Distribution (DIST)

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.

20010101

1-4-7

Distribution (DIST)

k Student-t Distribution

uStudent-t Probability Density

Student-t probability density calculates t probability density from a specified x value.

df+1

–

df + 1

x²

f (x) =

πdf

Perform the following key operations 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 V-

Window 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

200101101

1-4-8

Distribution (DIST)

uStudent-t Distribution Probability

Student-t distribution probability calculates the probability of t distribution data falling

between two specific values.

df + 1

df+1

a : lower boundary

b : upper boundary

–

x²

p =

∫

πdf

Perform the following key operations 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.

200101101

1-4-9

Distribution (DIST)

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 χ²Distribution

uχ Probability Density

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

specified x value.

x ^{2 –1}e^–

f(x) =

Perform the following key operations from the statistical data list.

5(DIST)

d(χ²)

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.

200101101

1-4-10

Distribution (DIST)

Calculation Result Output Example

p .................................. χ²probability density

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

# Current V-Window settings are used for

graph drawing when the SET UP screen's

[Stat Wind] setting is [Manual]. The V-

Window settings below are set automatically

Xmin = 0, Xmax = 11.5, Xscale = 2,

Ymin = -0.1, Ymax = 0.5, Yscale = 0.1

20010101

1-4-11

Distribution (DIST)

uχ Distribution Probability

χ²distribution probability calculates the probability of χ²distribution data falling between two

specific values.

a : lower boundary

b : upper boundary

x ^{2 –1}e^–dx

p =

∫

Perform the following key operations from the statistical data list.

5(DIST)

d(χ )

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 χ distribution

probability.

200101101

1-4-12

Distribution (DIST)

Calculation Result Output Example

p .................................. χ²distribution probability

k F Distribution

u F Probability Density

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

specified x value.

n + d

nx^–^{n + d}

–1

f (x) =

²x ²

1 +

Perform the following key operations 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.

200101101

1-4-13

Distribution (DIST)

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].

20010101

1-4-14

Distribution (DIST)

u F Distribution Probability

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

specific values.

a : lower boundary

b : upper boundary

n + d

–

–1

x ²

p =

1 +

∫

Perform the following key operations 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.

200101101

1-4-15

Distribution (DIST)

Calculation Result Output Example

p .................................. F distribution probability

20010101

1-4-16

Distribution (DIST)

k Binomial Distribution

u Binomial Probability

Binomial probability calculates a probability at a specified value for the discrete binomial

distribution with the specified number of trials and probability of success on each trial.

p^x(1–p)^{n – x}

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

(0 < p < 1)

f (x) =

n : number of trials

Perform the following key operations 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

(List 1 to 20)

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 a parameter data specification item that is 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.

200101101

1-4-17

Distribution (DIST)

Calculation Result Output Example

p .................................. binomial probability

uBinomial Cumulative Density

Binomial cumulative density calculates a cumulative probability at a specified value for the

discrete binomial distribution with the specified number of trials and probability of success on

each trial.

Perform the following key operations 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

(List 1 to 20)

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 a parameter data specification item that is different from

list data specification.

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

200101101

1-4-18

Distribution (DIST)

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 ......................................... probability of success

200101101

1-4-19

Distribution (DIST)

k Poisson Distribution

uPoisson Probability

Poisson probability calculates a probability at a specified value for the discrete Poisson

distribution with the specified mean.

e^–^µµ

f(x)=

(x = 0, 1, 2, ···)

µ : mean (µ > 0)

Perform the following key operations 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

(List 1 to 20)

µ .................................. 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 a 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.

200101101

1-4-20

Distribution (DIST)

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 operations 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

(List 1 to 20)

µ .................................. 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 a 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 probability

200101101

1-4-21

Distribution (DIST)

k Geometric Distribution

uGeometric Probability

Geometric probability calculates the probability at a specified value, and the number of the

trial on which the first success occurs, for the geometric distribution with a specified

probability of success.

f (x)= p(1– p)^{x – 1}

(x = 1, 2, 3, ···)

Perform the following key operations 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

(List 1 to 20)

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 a 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

# Positive integer number is calculated whether

list data (Data:List) or x value (Data:variable)

is specified.

# There is no graphing for geometric distribu-

tion.

200101101

1-4-22

Distribution (DIST)

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 operations 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

(List 1 to 20)

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 a 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 probability

# Positive integer number is calculated whether

list data (Data:List) or x value (Data:variable)

is specified.

200101101

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