Chapter 7, Problem 137E

To determine

To graph: The back-to-back stem plot to represent the school performance, on the basis of grade point average (GPA) and a standardized test (IQ) of 78 seventh-grade student.

Explanation of Solution

Graph: The back-to-back stem plot is useful to compare the two related distributions by having the same stem and two leaves.

For the grade point average (GPA) of 78 seventh-grade student, draw back-to-back stemplot. The values on the left side represent the women scores and the values on the right side represent the men score. The back-to-back stemplot for comparing the distributions is shown below:

For a standardized test (IQ) of 78 seventh-grade student, draw back-to-back stemplot. The values on the left side represent the women scores and the values on the right side represent the men score. The back-to-back stemplot for comparing the distributions is shown below:

Interpretation: From the above graphs, it can be concluded that the distribution for the grade point average (GPA) and a standardized test (IQ) of 78 seventh-grade student appears almost similar.

To determine

To test: The significance difference for the grade point average (GPA) and a standardized test (IQ) of 78 seventh-grade student on the basis of gender.

Answer to Problem 137E

Solution: The difference is insignificant for the grade point average (GPA) scores of men and women. For a standardized test (IQ), the difference is insignificant as the test results are strongly significant.

Explanation of Solution

Calculation: For the difference in grade point average (GPA), the testing can be done as shown below:

The null hypothesis assumes that on an average there is no significant difference on the grade point average (GPA) scores of men and women while the alternative hypothesis assumes that on an average the grade point average (GPA) scores of men is less than the women. Symbolically, the hypothesis can be represented as follows,

$\begin{array}{l}{H}_0:{\mu }_M={\mu }_F\\ H_a:{\mu }_M<{\mu }_F\end{array}$

Where, ${\mu }_M$ represent the men grade point averages and ${\mu }_F$ represent the women grade point averages.

To test the significant difference for the grade point averages, perform the following steps in Minitab,

Step 1: Enter the data in the worksheet of Minitab.

Step 2: Sort the data for ‘GPA’ on the basis of ‘Gender’. Here, ‘2’ represent the men and ‘1’ represent the women. Go to Data, click on sort, select ‘GPA’ in the ‘Sort column’ and enter ‘Gender’ in the ‘By column’ textbox. Also, store sorted data in the column of current worksheet. After this, make two columns, one for ‘GPA (M)’and other one for ‘GPA (F)’.

Step 3: In a Minitab worksheet go to ‘Stat’ point on ‘Basic Statistics’ and click on ‘2-Sample t’.

Step 4: In the dialogue box that appears select the samples in different column and enter the variable ‘GPA (M)’ in the first textbox and the variable ‘GPA (F)’ in the second textbox.

Step 5: Next click on options tab and set the confidence level as 95 and the Alternative hypothesis as ‘less than’. Finally click on OK twice to obtain the output.

From the Minitab results, the test statistic value is obtained as $\underset{\_}{-0.91}$, the degree of freedom is 74, and the P-value is obtained as 0.184.

For the difference in the standardized test (IQ) scores, the testing can be done as shown below:

The null hypothesis assumes that on an average there is no significant difference between the standardized test (IQ) scores of men and women while the alternative hypothesis assumes that on an average the standardized test (IQ) scores of men is greater than the women. Symbolically, the hypothesis can be represented as follows,

$\begin{array}{l}{H}_0:{\mu }_M={\mu }_F\\ H_a:{\mu }_M>{\mu }_F\end{array}$

Where, ${\mu }_M$ represent the men standardized test (IQ) scores and ${\mu }_F$ represent the women standardized test (IQ) scores.

To test the significant difference for the standardized test (IQ) scores, perform the following steps in Minitab,

Step 1: Enter the data in the worksheet of Minitab.

Step 2: Sort the data for ‘IQ’ on the basis of ‘Gender’. Here, ‘2’ represent the men and ‘1’ represent the women. Go to Data, click on sort, select ‘IQ’ in the ‘Sort column’ and enter ‘Gender’ in the ‘By column’ textbox. Also, store sorted data in the column of current worksheet. After this, make two columns, one for ‘IQ (M)’and other one for ‘IQ (F)’.

Step 3: In a Minitab worksheet go to ‘Stat’ point on ‘Basic Statistics’ and click on ‘2-Sample t’.

Step 4: In the dialogue box that appears select the samples in different column and enter the variable ‘IQ (M)’ in the first textbox and the variable ‘IQ (F)’ in the second textbox.

Step 5: Next click on options tab and set the confidence level as 95 and the Alternative hypothesis as ‘greater than’. Finally click on OK twice to obtain the output.

From the Minitab results, the test statistic value is obtained as $\underset{\_}{1.64}$, the degree of freedom is 56, and the P-value is obtained as 0.053.

Conclusion: Hence, for the grade point average (GPA), the P-value is greater than 0.05, so at 5% level of significance the null hypothesis will be accepted and it is concluded that the difference is insignificant. For the standardized test (IQ) scores, the P-value is greater than 0.05, so at 5% level of significance, the null hypothesis will be accepted and it is concluded that the difference is insignificant.

To determine

To find: The 95% confidence intervals for the grade point average (GPA) and the standardized test (IQ) scores of 78 seventh-grade student on the basis of gender.

Answer to Problem 137E

Solution: For the grade point average (GPA), the confidence interval is $\underset{\_}{\left(-1.328,0.498\right)}$ and for the standardized test (IQ), the confidence interval is $\underset{\_}{\left(-1.12,11.36\right)}$.

Explanation of Solution

Calculation: The 95% confidence intervals for the grade point average (GPA) is obtained by performing the following steps in Minitab,

Step 1: Enter the data in the worksheet of Minitab.

Step 2: Sort the data for ‘GPA’ on the basis of ‘Gender’. Here, ‘2’ represent the men and ‘1’ represent the women. Go to Data, click on sort, select ‘GPA’ in the ‘Sort column’ and enter ‘Gender’ in the ‘By column’ textbox. Also, store sorted data in the column of current worksheet. After this, make two columns, one for ‘GPA (M)’and other one for ‘GPA (F)’.

Step 3: In a Minitab worksheet go to ‘Stat’ point on ‘Basic Statistics’ and click on ‘2-Sample t’.

Step 4: In the dialogue box that appears select the samples in different column and enter the variable ‘GPA (M)’ in the first textbox and the variable ‘GPA (F)’ in the second textbox.

Step 5: Next click on options tab and set the confidence level as 95 and the Alternative hypothesis as ‘not equal to’. Finally click on OK twice to obtain the output.

Hence, the confidence interval is obtained as $\underset{\_}{\left(-1.328,0.498\right)}$.

The 95% confidence intervals for the standardized test (IQ) scores is obtained by performing the following steps in Minitab,

Step 1: Enter the data in the worksheet of Minitab.

Step 2: Sort the data for ‘IQ’ on the basis of ‘Gender’. Here, ‘2’ represent the men and ‘1’ represent the women. Go to Data, click on sort, select ‘IQ’ in the ‘Sort column’ and enter ‘Gender’ in the ‘By column’ textbox. Also, store sorted data in the column of current worksheet. After this, make two columns, one for ‘IQ (M)’and other one for ‘IQ (F)’.

Step 3: In a Minitab worksheet go to ‘Stat’ point on ‘Basic Statistics’ and click on ‘2-Sample t’.

Step 4: In the dialogue box that appears select the samples in different column and enter the variable ‘IQ (M)’ in the first textbox and the variable ‘IQ (F)’ in the second textbox.

Step 5: Next click on options tab and set the confidence level as 95 and the Alternative hypothesis as ‘not equal to’. Finally click on OK twice to obtain the output.

Hence, the confidence interval is obtained as $\underset{\_}{\left(-1.12,11.36\right)}$.

Interpretation: Therefore, it can be concluded that the 95% of the average standardized IQ scores lie between the values $\left(-1.12,11.36\right)$ and also, 95% of the average grade point average (GPA) lie between the values $\left(-1.328,0.498\right)$.

To determine

To explain: The findings of the graphical displays, significant test, and confidence intervals for the grade point average (GPA) and a standardized test (IQ) on the basis of gender.

Answer to Problem 137E

Solution: For the grade point average (GPA) and a standardized test (IQ), the results are not significant as the P-value in both the cases are greater than the level of significance 0.05. Also, the distribution appears to be similar in both the cases.

Explanation of Solution

For the grade point average (GPA), the test statistic is $-0.91$, degree of freedom is $74$, the P-value is $0.184$, and the confidence interval is $\left(-1.328,0.498\right)$. For the standardized test (IQ) the test statistic is $1.64$, degree of freedom is $56$, the P-value is $0.053$, and the confidence interval is $\left(-1.12,11.36\right)$. From the graphs obtained above, it can be said that the distribution are almost similar for the grade point average (GPA) and a standardized test (IQ).