User manual - Tests (TEST)

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2.Tests (TEST)

The Z Test provides a variety of different standardization-based tests. They make it possible

to test whether or not a sample accurately represents the population when the standard

deviation of a population (such as the entire population of a country) is known from previous

tests. Z testing is used for market research and public opinion research that need to be

performed repeatedly.

1-Sample Z Test tests for unknown population mean when the population standard

deviation is known.

2-Sample Z Test tests the equality of the means of two populations based on independent

samples when both population standard deviations are known.

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

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

The t Test tests the hypothesis when the population standard deviation is unknown. The

hypothesis that is the opposite of the hypothesis being proven is called the

null hypothesis

while the hypothesis being proved is called the

alternative hypothesis

. The t-test is normally

applied to test the null hypothesis. Then a determination is made whether the null hypothesis

or alternative hypothesis will be adopted.

1-Sample t Test tests the hypothesis for a single unknown population mean when the

population standard deviation is unknown.

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

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

Test tests hypothesis concerning the proportion of samples included in each of a number

of independent groups. Mainly, it generates cross-tabulation of two categorical variables

(such as yes, no) and evaluates the independence of these variables. It could be used, for

example, to evaluate the relationship between whether or not a driver has ever been

involved in a traffic accident and that person’s knowledge of traffic regulations.

2-Sample F Test tests the hypothesis for the ratio of sample variances. It could be used, for

example, to test the carcinogenic effects of multiple suspected factors such as tobacco use,

alcohol, vitamin deficiency, high coffee intake, inactivity, poor living habits, etc.

ANOVA tests the hypothesis that the population means of the samples are equal when there

are multiple samples. It could be used, for example, to test whether or not different combina-

tions of materials have an effect on the quality and life of a final product.

One-Way ANOVA is used when there is one independent variable and one dependent

variable.

Two-Way ANOVA is used when there here are two independent variables and one depend-

ent variable.

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

described above. Full details concerning statistical principles and terminology can be found

in any standard general statistics textbook.

Summary of content (25 pages)

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6 2.Tests (TEST) The Z Test provides a variety of different standardization-based tests. They make it possible to test whether or not a sample accurately represents the population when the standard deviation of a population (such as the entire population of a country) is known from previous tests. Z testing is used for market research and public opinion research that need to be performed repeatedly. 1-Sample Z Test tests for unknown population mean when the population standard deviation is known.
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7 On the initial STAT2 Mode screen, press 3 (TEST) to display the test menu, which contains the following items. • 3(TEST)b(Z) ... Z Tests (p.7) c(T) ... t Tests (p.15) d(χ2) ... χ2 Test (p.23) e(F) ... 2-Sample F Test (p.25) f(ANOVA) ... ANOVA (p.27) k Z Tests uZ Test Common Functions You can use the following graph analysis functions after drawing a graph. • 1(Z) ... Displays z score.
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8 Perform the following key operation from the statistical data list. 3(TEST) b(Z) b(1-Smpl) The following shows the meaning of each item in the case of list data specification. Data ............................ data type µ .................................. population mean value test conditions (“G µ0” specifies two-tail test, “< µ0” specifies lower one-tail test, “> µ0” specifies upper one-tail test.) µ0 ................................. assumed population mean σ ..................................
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9 • 1(CALC) ... Performs the calculation. • 6(DRAW) ... Draws the graph. Calculation Result Output Example µG11.4 ........................ direction of test z .................................. p .................................. o .................................. xσn-1 ............................. Z score p-value mean of sample 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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10 u2-Sample Z Test This test is used when the sample standard deviations for two populations are known to test the hypothesis. The 2-Sample Z Test is applied to normal distribution. Z= o1 – o2 σ12 σ22 n1 + n2 o1 : mean of sample 1 o2 : mean of sample 2 σ1 : population standard deviation of sample 1 σ2 : population standard deviation of sample 2 n1 : size of sample 1 n2 : size of sample 2 Perform the following key operation from the statistical data list.
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11 o1 ................................. n1 ................................. o2 ................................. n2 ................................. 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.
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12 u1-Prop Z Test This test is used to test for an unknown proportion of successes. The 1-Prop Z Test is applied to normal distribution. Z= x n – p0 p0 (1– p0) n p0 : expected sample proportion n : size of sample Perform the following key operation from the statistical data list. 3(TEST) b(Z) d(1-Prop) Prop ............................ sample proportion test conditions (“G p0” specifies two-tail test, “< p0” specifies lower one-tail test, “> p0” specifies upper one-tail test.) p0 ....................
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13 u2-Prop Z Test This test is used to compare the proportion of successes. The 2-Prop Z Test is applied to normal distribution. x1 x 2 n1 – n2 Z= p(1 – p ) 1 + 1 n1 n2 x1 : data value of sample 1 x2 : data value of sample 2 n1 : size of sample 1 n2 : size of sample 2 p̂ : estimated sample proportion Perform the following key operation from the statistical data list. 3(TEST) b(Z) e(2-Prop) p1 .................................
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14 p1>p2 ............................ z .................................. p .................................. p̂1 ................................. p̂2 ................................. p̂ .................................. n1 ................................. n2 .................................
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15 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.
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16 u1-Sample t Test This test uses the hypothesis test for a single unknown population mean when the population standard deviation is unknown. The 1-Sample t Test is applied to t-distribution. t= o – µ0 xσ n–1 n o : mean of sample µ0 : assumed population mean xσn-1 : sample standard deviation n : size of sample Perform the following key operation from the statistical data list. 3(TEST) c(T) b(1-Smpl) The following shows the meaning of each item in the case of list data specification. Data ...........
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17 Calculation Result Output Example µ G 11.3 ...................... direction of test t ................................... p .................................. o .................................. xσn-1 ............................. n .................................. t score p-value mean of sample Sample standard deviation size of sample # [Save Res] does not save the µ condition in line 2.
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18 u2-Sample t Test 2-Sample t Test compares the population means when standard deviations are unknown. The 2-Sample t Test is applied to t-distribution. t= o1 – o 2 x1σ n –12 x2σ n –12 n1 + n 2 o1 : mean of sample 1 o2 : mean of sample 2 x1σn-1 : standard deviation of sample 1 x2σn-1 : standard deviation of sample 2 n1 : size of sample 1 n2 : size of sample 2 This formula is applicable when the sample is not pooled, and the denominator is different when the sample is pooled.
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19 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(2) ..........................
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20 p .................................. df ................................. o1 ................................. o2 ................................. x1σn-1 ............................ x2σn-1 ............................ xpσn-1 ............................ p-value degrees of freedom mean of sample 1 mean of sample 2 standard deviation of sample 1 standard deviation of sample 2 pooled sample standard deviation (Displayed only when Pooled: On Setting.) n1 .................................
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21 uLinearReg t Test LinearReg t Test treats paired-variable data sets as (x, y) pairs and plots all data on a graph. Next, a straight line (y = a + bx) is drawn through the area where the greatest number of plots are located and the degree to which a relationship exists is calculated.
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22 Calculation Result Output Example β G 0 & ρ G 0 .............. direction of test t ................................... p .................................. df ................................. a .................................. b .................................. s .................................. r .................................. r2 .................................
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23 k χ2 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). n : all data values k expected counts Σ x ×Σ x = Σn ij Fij i =1 ij j =1 (xij – Fij)2 Fij i =1 j =1 k χ2 = Σ Σ Perform the following key operation from the statistical data list. 3(TEST) d(χ2) Next, specify the matrix that contains the data.
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24 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 χ2 ................................. χ2 value p .................................. p-value df ................................. degrees of freedom You can use the following graph analysis functions after drawing a graph. • 1(CHI) ...
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25 k 2-Sample F Test 2-Sample F Test tests the hypothesis for the ratio of sample variances. The F Test is applied to F distribution. F= x1σn–12 x2σn–12 Perform the following key operation from the statistical data list. 3(TEST) e(F) The following is the meaning of each item in the case of list data specification. Data ............................ data type σ1 .................................
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26 Calculation Result Output Example σ1Gσ2 .......................... direction of test F .................................. p .................................. o1 ................................. o2 ................................. x1σn-1 ............................ x2σn-1 ............................ n1 ................................. n2 .................................
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27 k ANOVA ANOVA tests the hypothesis that the population means of the samples are equal when there are multiple samples. One-Way ANOVA is used when there is one independent variable and one dependent variable. Two-Way ANOVA is used when there here are two independent variables and one dependent variable. This Two-Way ANOVA calculation is available under the condition that is to prepare more than two experimental data as each dependent data.
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28 Calculation Result Output Example One-Way ANOVA Line 1 (A) .................... Factor A df value, SS value, MS value, F value, p-value Line 2 (ERR) ............... Error df value, SS value, MS value Two-Way ANOVA Line 1 (A) .................... Factor A df value, SS value, MS value, F value, p-value Line 2 (B) .................... Factor B df value, SS value, MS value, F value, p-value Line 3 (AB) .................. Factor A x Factor B df value, SS value, MS value, F value, p-value Line 4 (ERR) ......
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29 k ANOVA (Two-Way) uDescription The nearby table shows measurement results for a metal product produced by a heat treatment process based on two treatment levels: time (A) and temperature (B). The experiments were repeated twice each under identical conditions. B (Heat Treatment Temperature) A (Time) B1 B2 A1 113 , 116 139 , 132 A2 133 , 131 126 , 122 Perform analysis of variance on the following null hypothesis, using a significance level of 5%.
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30 uInput Example uResults