Chapter 6, Problem 130E

Section 1

To determine

To find: The test statistic and the p- value for sample size 10.

Answer to Problem 130E

Solution: The value of Z test statistic is 1.58 and the p-value is 0.057.

Explanation of Solution

Calculation: To calculate the test statistic and the p-value follow the below mentioned steps in Minitab;

Step 1: Go to $\text{Stat}\to \text{Basic}\text{statistics}\to 1\text{-sample }Z$.

Step 2: In the dialog box that appears select ‘Summarized data’. Then enter 10 under the field marked as ‘Sample size’, 16 under the field marked as ‘Standard deviation’ and 8 under ‘Mean’ field.

Step 3: Click on ‘Perform Hypothesised test’. Enter 0 under the field marked as ‘Hypothesised mean’. Click on ‘option’.

Step 4: In the dialog box that appears, enter ‘95.0’ under the field marked as ‘Confidence level’ and select ‘greater than’ under the field marked as ‘Alternative hypothesis’. Finally, click on ‘OK’.

From Minitab results, The Z test statistic is 1.58 and the p-value is 0.057. Since the p-value is greater than significance level (0.05), the null hypothesis is accepted.

Section 2

To determine

To find: The test statistic and the p- value for sample size 20.

Answer to Problem 130E

Solution: The Z test statistic is 2.24 and the p-value 0.013.

Explanation of Solution

Calculation: To calculate the test statistic and the p-value follow the below mentioned steps in Minitab;

Step 1: Go to $\text{Stat}\to \text{Basic}\text{statistics}\to 1\text{-sample }Z$.

Step 2: In the dialog box that appears select ‘Summarized data’. Then enter 20 under the field marked as ‘Sample size’, 16 under the field marked as ‘Standard deviation’ and 8 under ‘Mean’ field.

Step 3: Click on ‘Perform Hypothesised test’. Enter 0 under the field marked as ‘Hypothesised mean’. Click on ‘option’.

Step 4: In the dialog box that appears, enter ‘95.0’ under the field marked as ‘Confidence level’ and select ‘greater than’ under the field marked as ‘Alternative hypothesis’. Finally, click on ‘OK’.

From Minitab results, The Z test statistic is 2.24 and the p-value is 0.013. Since the p-value is less than significance level (0.05), the null hypothesis is rejected.

Section 3

To determine

To find: The test statistic and the p- value for sample size 30.

Answer to Problem 130E

Solution: The Z test statistic is 2.74 and the p-value is 0.003.

Explanation of Solution

Calculation: To calculate the test statistic and the p-value follow the below mentioned steps in Minitab;

Step 1: Go to $\text{Stat}\to \text{Basic}\text{statistics}\to 1\text{-sample }Z$.

Step 2: In the dialog box that appears select ‘Summarized data’. Then enter 30 under the field marked as ‘Sample size’, 16 under the field marked as ‘Standard deviation’ and 8 under ‘Mean’ field.

Step 3: Click on ‘Perform Hypothesised test’. Enter 0 under the field marked as ‘Hypothesised mean’. Click on ‘option’.

Step 4: In the dialog box that appears, enter ‘95.0’ under the field marked as ‘Confidence level’ and select ‘greater than’ under the field marked as ‘Alternative hypothesis’. Finally, click on ‘OK’.

From Minitab results, The Z test statistic is 2.74 and the p-value is 0.003. Since the p-value is less than significance level (0.05), the null hypothesis is rejected.

Section 4

To determine

To find: The test statistic and the p- value for sample size 40.

Answer to Problem 130E

Solution: The Z test statistic is 3.16 and the p-value is 0.001.

Explanation of Solution

Calculation: To calculate the test statistic and the p-value follow the below mentioned steps in Minitab;

Step 1: Go to $\text{Stat}\to \text{Basic}\text{statistics}\to 1\text{-sample }Z$.

Step 2: In the dialog box that appears select ‘Summarized data’. Then enter 40 under the field marked as ‘Sample size’, 16 under the field marked as ‘Standard deviation’ and 8 under ‘Mean’ field.

Step 3: Click on ‘Perform Hypothesised test’. Enter 0 under the field marked as ‘Hypothesised mean’. Click on ‘option’.

Step 4: In the dialog box that appears, enter ‘95.0’ under the field marked as ‘Confidence level’ and select ‘greater than’ under the field marked as ‘Alternative hypothesis’. Finally, click on ‘OK’.

From Minitab results, The Z test statistic is 3.16 and the p-value is 0.001. Since the p-value is less than significance level (0.05), the null hypothesis is rejected.

Section 5

To determine

To find: The test statistic and the p- value for sample size 50.

Answer to Problem 130E

Solution: The Z test statistic is 3.54 and the p-value is0.000.

Explanation of Solution

To calculate the test statistic and the p-value follow the below mentioned steps in Minitab;

Step 1: Go to $\text{Stat}\to \text{Basic}\text{statistics}\to 1\text{-sample }Z$.

Step 2: In the dialog box that appears select ‘Summarized data’. Then enter 50 under the field marked as ‘Sample size’, 16 under the field marked as ‘Standard deviation’ and 8 under ‘Mean’ field.

Step 3: Click on ‘Perform Hypothesised test’. Enter 0 under the field marked as ‘Hypothesised mean’. Click on ‘option’.

Step 4: In the dialog box that appears, enter ‘95.0’ under the field marked as ‘Confidence level’ and select ‘greater than’ under the field marked as ‘Alternative hypothesis’. Finally, click on ‘OK’.

From Minitab results, The Z test statistic is 3.54 and the p-value is 0.000. Since the p-value is less than significance level (0.05), the null hypothesis is rejected.

Section 6

To determine

To graph: The values of test statistic versus sample size using the results of sections (1), (2), (3), (4) and (5).

Explanation of Solution

Graph: From the results of section 1 to 5, create a table for Sample size and test statistic.

Sample size	Test statistic
10	1.58
20	2.24
30	2.74
40	3.16
50	3.54

To plot the graph, follow the below mentioned steps in Minitab;

Step1: Enter the data of table in Minitab. Enter variable name as Sample size and test statistic.

Step 2: Go to $\text{Graph}\to \text{Scatter}\text{plot}$

Step 3: In the dialog box select ‘simple’ then click Ok.

Step 4: In the new dialog box, enter ‘Test statistic’ as ‘Y variables’ ‘Sample size’ as ‘X variables’. Finally, click OK.

From the Minitab results, the graph is as follows,

Section 7:

To determine

To graph: The values of p-test versus sample size using the results of sections (1), (2), (3), (4), and (5).

Explanation of Solution

Graph: From the results of section 1 to 5, create a table for Sample size and test statistic.

Sample size	p-value
10	0.057
20	0.013
30	0.003
40	0.001
50	0.000

To plot the graph, follow the below mentioned steps in Minitab;

Step1: Enter the data of table in Minitab. Enter variable name as Sample size and p-value.

Step 2: Go to $\text{Graph}\to \text{Scatter}\text{plot}$

Step 3: In the dialog box select ‘simple’ then click Ok.

Step 4: In the new dialog box, enter ‘p-value’ under the field marked as ‘Y variables’ ‘Sample size’ under the field marked as ‘X variable’. Finally, click OK.

From the Minitab results, the graph is as follows,

To determine

To explain: The effect of the increment in sample size on test statistic and p- value.

Answer to Problem 130E

Solution: As the sample size increases, the value of test statistic increases and the p- value decrease.

Explanation of Solution

From the scatter plots drawn above, it can be seen that the test statistic and the sample size are positively correlated whereas the p- value is negatively correlated.