Question

The American Association of University Professors (AAUP) conducts salary studies of college professors and publishes its findings, in AAUP Annual Report on the Economic Status of the Profession. In Example \(10.3\) on pages \(409-411\), we performed a hypothesis test based on independent samples to decide whether mean salaries differ for faculty in private and public institutions. Now you are to perform that same hypothesis test based on a paired sample. Pairs were formed by matching faculty in private and public institutions by rank and specialty. A random sample of \(30\) pairs yielded the data, in thousands of dollars, presented on the WeissStats site. Use the technology of your choice to do the following tasks.

a. Decide at the \(5%\) significance level, whether the data provide sufficient evidence to conclude that mean salaries differ for faculty in private institution and public institutions. Use the paired \(t-\)test\).

b. Compare your results in part (a) to the one obtained in Example \(10.3\).

c. Repeat both the pooled \(t-\)test of Example \(10.3\) and the paired \(t-\)test of part (a) using a \(1%\) significance level and compare your results.

d. Which test do you think is preferable here: the pooled \(t-\)test or the paired \(t-\)test? Explain your answer.

e. Find and interpret a \(95%\) confidence interval for the difference between the mean salaries of faculty in private and public institution. Use the paired \(t-\)interval procedure.

f. Compare your results in part (e) to the one obtained in Example \(10.4\) on page \(412\).

g. Obtain a normal probability plot and a boxplot of the paired differences.

h. Based on your graphs from part (g), do you think that applying paired \(t-\)procedures here is reasonable?

Short answer

Part a. The data provide sufficient evidence to conclude that mean salaries differ for faculty in private institutions and public institutions at \(5%\) level.

Part b. Both conclusions are same.

Part c. The both conclusions are same.

Part d. The Paired t-test would be preferable.

Part e. The \(95%\) confidence interval is \(12.65, 17.41\)

Part f. The confidence interval obtained in Example 10.4 is wider when compared to paired \(t-\)test interval.

Part g. The normal probability plot, it is observed that the distribution of data is normally distributed because the observations are closer to straight line.

The box plot distribution of data is roughly symmetric. That is shape of the distribution, is bell shaped.

Part h. Yes, it is reasonable to applying paired t-procedures.

Step-by-step solution

Part a. Step 1. Given information

\(N=30\)

\(\alpha =5%\)

Part a. Step 2. Calculation

The Null hypothesis:

\(H_{0}:\mu_{1}=\mu_{2}\)

That is the mean salaries do not differ for faculty in private institutions and public institutions.

The alternative hypothesis:

\(H_{a}:\mu_{1}\neq \mu_{2}\)

That is the mean salaries differ for faculty in private institutions and public institutions.

Using MINITAB the value of test statistic and p-value is,

From the MINITAB output, the value of test statistic is,

test statistic \(=12.91\)

\(P-value=0\)

If \(P-value \leq \alpha\), then reject the null hypothesis.

\(P-value \leq \alpha\)

\(0\leq 0.05\)

Here, the \(P-\)value is \(0\) which is lesser than the level of significance. Therefore, the null hypothesis is rejected at \(5%\) level significance.

Thus, it can be conclude that the test results are statistically significant at \(5%\) level of significance and data provide sufficient evidence to conclude that mean salaries differ for faculty in private institutions and public institutions at \(5%\) level.

Part b. Step 1. Calculation

Example \(10.3\): Pooled t-test.

Here, the null hypothesis is rejected at \(5%\) level of significance. This indicates that the results are statistically significant. Therefore, the mean salaries of faculty in private institutions and public institutions are different.

Thus, the data provide sufficient evidence to conclude that mean salaries for faculty in private and public institutions differ.

Part (a).Paired t-test.

Here, the null hypothesis is rejected at \(5%\) level. Thus, it can be conclude that the test results are statistically significant at \(5%\) level of significance.

Thus, the data provide sufficient evidence to conclude that mean salaries differ for faculty in private institutions and public institutions at \(5%\) level.

Thus, the both conclusions are same.

Part c. Step 1. Given information

\(N=30\)

\(\alpha=1%\)

Part c. Step 2. Calculation

The Null hypothesis:

\(H_{0}:\mu_{1}=\mu_{2}\)

That is the mean salaries do not differ for faculty in private institutions and public institutions.

The alternative hypothesis:

\(H_{a}:\mu_{1}\neq\mu_{2}\)

That is the mean salaries differ for faculty in private institutions and public institutions.

Using MINITAB, obtain the value of test statistic and p-value.

From the MINITAB output, the value of

test statistic\(=12.91\)

\(P-value=0\)

If \(P-\)value \(\leq \alpha\), then reject the null hypothesis

\(P-\)value \(\leq \alpha\)

\(0\leq 0.01\)

Here, the \(P-\)value is \(0\) which is lesser than the level of significance. Therefore, the null hypothesis is rejected at \(1%\) level significance.

Thus, it can be conclude that the test results are statistically significant at \(1%\) level of significance and data provide sufficient evidence to conclude that mean salaries differ for faculty in private institutions and public institutions at \(1%\) level.

Pooled t-test:

Using MINITAB, obtain the value of test statistic and p-value.

From the MINITAB output is,

test statistic\(=2.32\)

\(P-value=0.024\)

If \(P-\)value \(\leq \alpha\) , then reject the null hypothesis otherwise accept the null hypothesis.

\(P-\)value \(\leq \alpha\)

\(0.024>0.01\)

Here, the \(P-\)value is \(0.024\) which is greater than the level of significance. That is \(P(=0.024)> \alpha(=0.01)\) therefore, the null hypothesis is not rejected at \(1%\) level.

Thus, it can be conclude that the test results are not statistically significant at \(1%\) level of significance.

Comparison:

Paired t-test:

Here, the null hypothesis is rejected at \(1%\) level. Thus, it can be conclude that the test results are statistically significant at \(1%\) level of significance.

Thus, the data provide sufficient evidence to conclude that mean salaries differ for faculty in private institutions and public institutions at \(1%\) level.

Pooled t-test:

Here, the null hypothesis is not rejected at \(1%\) level of significance. This indicates that the results are not statistically significant. Therefore, the mean salaries of faculty in private institutions and public institutions are not different.

Thus, the data do not provide sufficient evidence to conclude that mean salaries for faculty in private and public institutions differ.

Thus, the both conclusions are not similar.

Part d. Step 1. Calculation

From the results, it is observed that the paired \(t-\)test would be preferable because there is a strong evidence against the null hypothesis at both level of significant compared to pooled \(t-\)test.

Thus, Paired \(t-\)test would be preferable.

Part e. Step 1. Calculation

UsingMINITAB outputin part (a).

The \(95%\) confidence interval is \((12.65, 17.41)\).

Part f. Step 1. Calculation

Comparison:

Example \(10.4\):

The \(95%\) confidence interval for the difference between the mean salaries of faculty in private and public institutions lies between \($2.49\) and \($27.53\) thousand.

Part (e):

The \(95%\) confidence interval for the difference between the mean salaries of faculty in private and public institutions lies between \($12.65\) and \($17.41\) thousand.

Thus, the confidence interval obtained in Example \(10.4\) is wider when compared to paired t-test interval.

Part g. Step 1. Calculation

By using MINITAB fornormal probability plot:

The normal probability plot, it is observed that the distribution of data is normally distributed because the observations are closer to straight line.

By using MINITAB for Boxplot:

The boxplot distribution of data is roughly symmetric. That is shape of the distribution, is bell shaped.

Part h. Step 1. Calculation

Yes, it is reasonable to applying paired t-procedures because the distribution of "paired difference” is approximately normally distributed.