Question

According to the article "Workaholism in Organizations: Gender Differences" (Sex Roles [1999]: \(333-346\) ), the following data were reported on 1996 income for random samples of male and female MBA graduates from a certain Canadian business school: \begin{tabular}{lccc} & \(\boldsymbol{N}\) & \(\overline{\boldsymbol{x}}\) & \(\boldsymbol{s}\) \\ \hline Males & 258 & \(\$ 133,442\) & \(\$ 131,090\) \\ Females & 233 & \(\$ 105,156\) & \(\$ 98,525\) \\ \hline \end{tabular} Note: These salary figures are in Canadian dollars. a. Test the hypothesis that the mean salary of male MBA graduates from this school was in excess of \(\$ 100,000\) in \(1996 .\) b. Is there convincing evidence that the mean salary for all female MBA graduates is above \(\$ 100,000 ?\) Test using \(\alpha=.10\) c. If a significance level of \(.05\) or \(.01\) were used instead of \(.10\) in the test of Part (b), would you still reach the same conclusion? Explain.

Short answer

a) Yes, the mean salary of male MBA graduates was significantly more than \$100,000 in 1996. b) Based on the significance level \(\alpha = 0.10\), the mean salary for female MBA graduates was not significantly above \$100,000 in 1996. c) With a significance level of \(0.05\) or \(0.01\), the decision might change, depending on the value of the test statistic.

Step-by-step solution

Hypothesis Formulation for Male Graduates

Formulate the null and alternative hypothesis. Here, we want to test whether the mean salary of male MBA graduates was more than \$100,000. The null hypothesis (\(H_0\)) assumes that the mean is \$100,000, and the alternative hypothesis (\(H_a\)) is that the mean is more than \$100,000. In symbols, this is: \(H_0: \mu = \$100,000\), \(H_a: \mu > \$100,000\).

Test Statistic Calculation for Male Graduates

Calculate the test statistic using the formula \(Z = (\overline{x} - \mu_0) / (s/ \sqrt{N})\) where \(\overline{x}\) is the sample mean, \(\mu_0\) is the hypothesized population mean from \(H_0\), \(s\) is the standard deviation and \(N\) is the sample size. Substitute with the provided values into the formula.

Decision for Male Graduates

Compare the calculated test statistic with the critical value from the standard normal distribution table (Z-table) for a one-tailed test. If the test statistic is greater than the critical value, we reject the null hypothesis. Since we don't have a significance level, we use \(0.05\) by default.

Hypothesis Formulation for Female Graduates

Formulate the null and alternative hypothesis for the female graduates. Here, we want to test whether the mean salary of female MBA graduates was more than \$100,000. The null hypothesis (\(H_0\)) assumes that the mean is \$100,000, and the alternative hypothesis (\(H_a\)) is that the mean is more than \$100,000. In symbols, this is: \(H_0: \mu = \$100,000\), \(H_a: \mu > \$100,000\).

Test Statistic Calculation for Female Graduates

Calculate the test statistic using the same formula as in Step 2 and insert the appropriate values for female graduates.

Decision for Female Graduates

Compare the calculated test statistic with the critical value at \(\alpha = 0.10\), since a significance level of \(0.10\) was provided for the female graduates. If the test statistic is greater than the critical value, we reject the null hypothesis.

Significance Level Change

If a different significance level such as \(0.05\) or \(0.01\) was used instead of \(0.10\), the critical value will be different. Compare the calculated test statistic with the new critical values. If it is greater than the critical value at \(0.01\) or \(0.05\), we would still reject the null hypothesis; otherwise, we won't.

Key concepts

Statistical Hypothesis Formulation

When we talk about statistical hypothesis formulation, it's all about setting up a starting assumption (called the null hypothesis) and an alternative one to test against it. In the context of income analysis for MBA graduates, the null hypothesis might state that the average income is a certain amount, say \$100,000. The alternative hypothesis would suggest that the actual average income is different from this amount, for example, it could be higher. This is represented symbolically as:

• Null Hypothesis (\(H_0\)): \(\mu = \$100,000\)
• Alternative Hypothesis (\(H_a\)): \(\mu > \$100,000\) (for testing if the income is greater).

It's crucial to define these hypotheses clearly, as they set the stage for the entire testing process.

Test Statistic Calculation

Calculating the test statistic is a central part of any hypothesis test. The test statistic calculation helps us determine how far our sample statistic is from the null hypothesis's proposed value, measured in standard error units. You do this by using a formula that incorporates the sample mean, hypothesized mean, sample standard deviation, and sample size. In mathematical terms, the formula is: \(Z = (\overline{x} - \mu_0) / (s/ \sqrt{N})\) where \(\overline{x}\) is the sample mean, \(\mu_0\) is the hypothesized mean, \(s\) is the sample standard deviation, and \(N\) is the sample size. This Z-score tells you how many standard deviations away from the hypothesized mean the sample mean lies.

Significance Level

The significance level, denoted by \(\alpha\), is the threshold for deciding whether or not to reject the null hypothesis. It represents the probability of rejecting the null hypothesis when it is in fact true, which is a type I error. Common choices for \(\alpha\) are 0.01, 0.05, and 0.10. Choosing a lower \(\alpha\) means being more conservative about claiming a result as significant — that is, requiring stronger evidence to reject the null hypothesis. When analyzing the MBA graduates’ incomes, setting different significance levels, such as 0.05 or 0.01 instead of 0.10, would require a larger test statistic to reject the null hypothesis and claim the incomes are significantly different from the hypothesized value.

MBA Graduates Income Analysis

Performing an income analysis on MBA graduates helps us understand the financial returns of obtaining an MBA degree. This kind of analysis often involves hypothesis testing to determine whether the average income of MBA graduates exceeds a certain benchmark. In our case, we are comparing it to a benchmark of \$100,000. This involves using sample data to make inferences about the population as a whole, considering factors such as mean income (\(\overline{x}\)) and variability (standard deviation, \(s\)). This analysis can help educational institutions and students make informed decisions about the value of an MBA program.

Gender Differences in Income

Investigating gender differences in income, especially among MBA graduates, involves exploring whether a gender pay gap exists and its extent. In our example, we compare the incomes of male and female MBA graduates using hypothesis testing. Recognizing potential differences can provide insights into the impact of gender on career progression and pay. It's not just about identifying the presence of a gap but also understanding the magnitude and causes to inform policies for achieving gender equity in the workplace.