Casio ALGEBRA FX 1 0 PLUS User Manual

ALGEBRA FX 2.0 PLUS  
FX 1.0 PLUS  
User’s Guide  
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(Additional Functions)  
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ALGEBRA FX 2.0 PLUS  
FX 1.0 PLUS  
(Additional Functions)  
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20010101  
1
Contents  
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  
1
Advanced Statistics Application  
1-1 Advanced Statistics (STAT)  
1-2 Tests (TEST)  
1-3 Confidence Interval (INTR)  
1-4 Distribution (DIST)  
20010101  
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.  
n
1
(yi – (axi+ b))2  
• Linear Regression ...  
• Quadratic Regression ...  
• Cubic Regression ...  
• Quartic Regression ...  
MSE =  
MSE =  
MSE =  
MSE =  
Σ
n 2  
i=1  
n
1
(yi – (ax 2  
i
+ bxi+ c))2  
+ bx 2+ cxi +d ))2  
(yi – (ax4+ bx 3 + cx 2 + dxi + e))2  
Σ
n 3  
i=1  
n
1
(yi – (ax3  
i
i
Σ
n 4  
i=1  
n
1
i
i
i
Σ
n 5  
i=1  
20010101  
1-1-2  
Advanced Statistics (STAT)  
n
1
(yi – (a + b ln xi ))2  
• Logarithmic Regression ...  
MSE =  
MSE =  
MSE =  
MSE =  
MSE =  
Σ
n 2  
i=1  
n
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 ...  
Σ
n 2  
i=1  
n
1
Σ
n 2  
i=1  
n
1
Σ
n 2  
i=1  
n
2
1
C
• Logistic Regression ...  
yi  
Σ
1 + ae-bx  
i
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.  
20011101  
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.  
20010101  
1-1-4  
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(GT) after drawing a graph switches to the parameter setting screen  
(GT function). Pressing u 5(GT) again returns to the graph screen.  
· The GT 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.  
20011101  
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.  
2
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.  
20011101  
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) ... χ2 Test (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 =  
σ
µo  
n
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  
20011101  
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.  
20011101  
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.  
20010101  
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.  
o
o
σ
σ
n
n
1
2
: 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 =  
σ12 σ22  
1
+
n1 n2  
2
1
2
: 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.)  
σ
σ
1
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.  
20011101  
1-2-6  
Tests (TEST)  
o
1
................................. mean of sample 1  
n
1
................................. size (positive integer) of sample 1  
................................. mean of sample 2  
o
2
n
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  
µ
1G  
µ2  
........................... direction of test  
z ...................................z score  
p .................................. p-value  
o
o
x
1
................................. mean of sample 1  
................................. mean of sample 2  
2
1σn-1 ............................ standard deviation of sample 1  
(Displayed only for Data: List setting.)  
x
2σn-1 ............................ standard deviation of sample 2  
(Displayed only for Data: List setting.)  
n
n
1
2
................................. size of sample 1  
................................. size of sample 2  
# [Save Res] does not save the µ1 condition in  
line 2.  
20010101  
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.  
p
0
: expected sample proportion  
x
n
p0  
n : size of sample  
Z =  
p0(1– p0)  
n
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.)  
p
0
................................. 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.  
20011101  
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.  
x1 x2  
x
x
n
n
1
2
: data value of sample 1  
: data value of sample 2  
: size of sample 1  
n1 n2  
Z =  
1
2
1
1
p(1– p )  
+
: size of sample 2  
n1 n2  
ˆp : estimated sample proportion  
Perform the following key operation from the statistical data list.  
3(TEST)  
b(Z)  
e(2-Prop)  
p
1
................................. 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.)  
x
1
................................. 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  
n
1
x
2
n
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  
200111011  
1-2-9  
Tests (TEST)  
p
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  
n
n
1
................................. size of sample 1  
................................. size of sample 2  
2
# [Save Res] does not save the p1 condition in  
line 2.  
20010101  
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.  
20010101  
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  
o
: mean of sample  
: assumed population mean  
t =  
µ
0
x
σn–1  
n
xσn-1 : sample standard deviation  
: size of sample  
n
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.  
20011101  
1-2-12  
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.  
20010101  
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.  
o
o
1
2
:mean of sample 1  
:mean of sample 2  
o1 o2  
t
=
1
1
2
x
x
1σn-1 :standard deviation of  
sample 1  
2σn-1 :standard deviation of  
sample 2  
n
n
+
xpσn–1  
n1 n2  
2
2
(n1–1)x1σ n–1 +(n2–1)x2σn–1  
xpσn–1  
=
n1 + n2 – 2  
1
:size of sample 1  
:size of sample 2  
2
df = n1 + n2 – 2  
x
pσn-1 :pooled sample standard  
deviation  
df :degrees of freedom  
The following applies when pooling is not in effect.  
o
o
1
2
:mean of sample 1  
:mean of sample 2  
o1 o2  
t =  
2
2
x1σn–1 x2σn–1  
x
x
1σn-1 :standard deviation of  
sample 1  
2σn-1 :standard deviation of  
sample 2  
n
n
+
n1  
n2  
1
df =  
C2 (1–C )2  
1
:size of sample 1  
:size of sample 2  
+
n1–1 n2–1  
2
df :degrees of freedom  
2
x1σn–1  
n1  
C =  
2
+
2
x1σn–1 x2σn–1  
n1  
n2  
Perform the following key operations from the statistical data list.  
3(TEST)  
c(T)  
c(2-Smpl)  
20011101  
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.  
o
1
................................. mean of sample 1  
x
1σn-1 ............................ standard deviation (x1σn-1 > 0) of sample 1  
n
1
................................. size (positive integer) of sample 1  
................................. mean of sample 2  
o
2
x
2σn-1 ............................ standard deviation (x2σn-1 > 0) of sample 2  
................................. size (positive integer) of sample 2  
n
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.  
20011101  
1-2-15  
Tests (TEST)  
Calculation Result Output Example  
µ1Gµ2 ........................... direction of test  
t ................................... t score  
p .................................. p-value  
df ................................. degrees of freedom  
o
o
x
x
x
1
................................. mean of sample 1  
................................. mean of sample 2  
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.)  
n
n
1
2
................................. size of sample 1  
................................. size of sample 2  
# [Save Res] does not save the µ1 condition in  
line 2.  
20010101  
1-2-16  
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.  
n
a
b
n
r
: intercept  
(x o)(y p)  
Σ
: 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  
Σ
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.  
20011101  
1-2-17  
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)  
2
k χ Test  
χ2 Test sets up a number of independent groups and tests hypothesis related to the  
proportion of the sample included in each group. The χ2 Test is applied to dichotomous  
variables (variable with two possible values, such as yes/no).  
k
Expected counts  
×
xij  
xij  
Σ Σ  
i=1  
j=1  
Fij =  
k
xij  
ΣΣ  
i=1 j=1  
(xij Fij)2  
k
χ2 =  
ΣΣ  
F
ij  
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.  
20011101  
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  
2
2
χ ................................. χ value  
p .................................. p-value  
df ................................. degrees of freedom  
You can use the following graph analysis functions after drawing a graph.  
2
1(CHI) ... Displays χ value.  
2
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 χ2 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 χ2 and 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.  
2
x1σn–1  
F =  
2
x2σn–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  
σ
1
................................. 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.  
x
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  
n
1
x
n
2
................................. size (positive integer) of sample 2  
20011101  
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  
σ
1Gσ2 .......................... direction of test  
F.................................. F value  
p .................................. p-value  
o
o
x
x
1
................................. mean of sample 1 (Displayed only for Data: List setting.)  
................................. mean of sample 2 (Displayed only for Data: List setting.)  
2
1σn-1 ............................ standard deviation of sample 1  
2σn-1 ............................ standard deviation of sample 2  
n
n
1
................................. size of sample 1  
................................. size of sample 2  
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  
20011101  
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)  
B1  
B2  
A1  
A2  
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.  
20011101  
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.  
20011101  
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.  
α
2
σ
n
Left = o Z  
α
2
σ
n
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)  
20011101  
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.  
σ12 σ22  
+
o
o
σ
1
2
: mean of sample 1  
: mean of sample 2  
: population standard deviation  
of sample 1  
: population standard deviation  
of sample 2  
α
Left = (o1 o2) Z  
n1 n2  
2
1
σ12 σ22  
α
2
+
Right = (o1 o2) + Z  
n1 n2  
σ
2
n
n
1
2
: 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)  
20011101  
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)  
σ
σ
1
................................. population standard deviation of sample 1 (σ1 > 0)  
2
................................. 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.  
o
1
................................. mean of sample 1  
n
1
................................. size (positive integer) of sample 1  
................................. mean of sample 2  
o
2
n
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)  
o
o
x
1
................................. mean of sample 1  
................................. mean of sample 2  
2
1σn-1 ............................ standard deviation of sample 1  
(Displayed only for Data: List setting.)  
x
2σn-1 ............................ standard deviation of sample 2  
(Displayed only for Data: List setting.)  
n
n
1
2
................................. size of sample 1  
................................. size of sample 2  
20011101  
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  
x
1 x  
x
n
α
2
Left = – Z  
1–  
n
n n  
1 x  
x
n
x
α
2
1–  
Right = + Z  
n 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  
20011101  
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.  
n
x
1, n2 : size of sample  
1, x2 : data  
x
x1  
x
x2  
n2  
1 1–  
2 1–  
n1  
n1 n2  
x1  
n1  
x2  
n2  
α
2
+
Left =  
Z  
n1  
n2  
x
x1  
x
x2  
n2  
1 1–  
2 1–  
n1  
n1 n2  
x1  
n1  
x2  
n2  
α
2
Right =  
+Z  
+
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)  
x
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  
n
1
x
2
n
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  
20011101  
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  
n
n
1
................................. size of sample 1  
................................. size of sample 2  
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.  
x
σn–1  
n
α
2
Left = otn 1  
x
σn–1  
n
α
2
Right = o+ tn 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.  
20011101  
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.  
α
2
1
1
2
+
Left = (o o )tn +n  
xpσn–1  
2 –2  
1
2
1
n1 n2  
α
2
1
1
2
Right = (o o )+ tn +n  
xpσn–1  
+
2 –2  
1
2
1
n1 n2  
2
2
(n1–1)x1σ n–1 +(n2–1)x2σn–1  
xpσn–1  
=
n1 + n2 – 2  
20011101  
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.  
2
2
x1σn–1 x2σn–1  
α
2
+
Left = (o1 o2)– tdf  
n1  
n2  
2
2
x1σn–1 x2σn–1  
α
2
Right = (o1 o2)+ tdf  
+
n1  
n2  
1
df =  
(1–C)2  
n2–1  
C2  
n1–1  
+
2
x1σn–1  
n1  
C =  
2
2
x1σn–1 x2σn–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.  
20011101  
1-3-11  
Confidence Interval (INTR)  
o
1
................................. mean of sample 1  
x
1σn-1 ............................ standard deviation (x1σn-1 > 0) of sample 1  
n
1
................................. size (positive integer) of sample 1  
................................. mean of sample 2  
o
2
x
2σn-1 ............................ standard deviation (x2σn-1 > 0) of sample 2  
................................. size (positive integer) of sample 2  
n
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  
o
o
x
x
x
1
................................. mean of sample 1  
................................. mean of sample 2  
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.)  
n
n
1
2
................................. 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.  
2
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)  
2
2
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
µ)2  
1
(σ > 0)  
2
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].  
20011101  
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  
µ
b e(x dx  
1
2
σ 2  
p =  
a
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.  
20011101  
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  
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 = ?  
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  
20011101  
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  
2
df + 1  
2
x2  
df  
1+  
Γ
f (x) =  
πdf  
df  
2
Γ
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  
20011101  
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  
2
df+1  
2
dx  
a : lower boundary  
b : upper boundary  
b
Γ
x2  
df  
p =  
1+  
df  
2
a
πdf  
Γ
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.  
20011101  
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 χ2 Distribution  
2
uχ Probability Density  
2
2
χ probability density calculates the probability density function for the χ distribution at a  
specified x value.  
df  
2
df  
x
2
x 2 –1e–  
1
df  
2
1
2
f(x) =  
Γ
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.  
20011101  
1-4-10  
Distribution (DIST)  
Calculation Result Output Example  
p .................................. χ2 probability 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)  
2
uχ Distribution Probability  
χ2 distribution probability calculates the probability of χ2 distribution data falling between two  
specific values.  
df  
2
b
a : lower boundary  
b : upper boundary  
df  
x
2
x 2 –1edx  
1
df  
2
1
2
p =  
a
Γ
Perform the following key operations from the statistical data list.  
5(DIST)  
2
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.  
2
# There is no graphing for χ distribution  
probability.  
20011101  
1-4-12  
Distribution (DIST)  
Calculation Result Output Example  
p .................................. χ2 distribution 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  
2
nxn + d  
d
n
n
Γ
1  
2
n
d
f (x) =  
2 x 2  
1 +  
n
2
d
2
Γ
Γ
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.  
20011101  
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  
2
n
2
n + d  
2
b
Γ
n
–1  
n
d
nx  
d
x 2  
p =  
1 +  
dx  
n
2
d
2
a
Γ
Γ
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.  
20011101  
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.  
px(1–p)n x  
(x = 0, 1, ·······, n) p : success probability  
(0 < p < 1)  
f (x) =  
n
C
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.  
20011101  
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  
20011101  
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  
20011101  
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.  
eµµ  
x!  
x
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.  
20011101  
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  
20011101  
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.  
20011101  
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.  
20011101  

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