Question

Gender Bias In a study \(^{52}\) examining gender bias, a nationwide sample of 127 science professors evaluated the application materials of an undergraduate student who had ostensibly applied for a laboratory manager position. All participants received the same materials, which were randomly assigned either the name of a male \(\left(n_{m}=63\right)\) or the name of a female \(\left(n_{f}=64\right) .\) Participants believed that they were giving feedback to the applicant, including what salary could be expected. The average salary recommended for the male applicant was \(\$ 30,238\) with a standard deviation of \(\$ 5152\) while the average salary recommended for the (identical) female applicant was \(\$ 26,508\) with a standard deviation of \(\$ 7348\). Does this provide evidence of a gender bias, in which applicants with male names are given higher recommended salaries than applicants with female names? Show all details of the test.

Short answer

The final answer depends on the calculated p-value based on the provided data. If the p-value is less than 0.05, that suggests there is a significant difference in the recommended salaries for male and female candidates, indicating gender bias. However, if the p-value is greater than 0.05, there is insufficient evidence to claim gender bias in the salary recommendations.

Step-by-step solution

State the Hypotheses

The null hypothesis (\(H_0\)) is that there is no gender bias, which means the mean salary of the male applicant equals the mean salary of female applicant, i.e., \(\mu_m = \mu_f\). The alternative hypothesis (\(H_a\)) is that there is a gender bias, which means the mean salary of male applicant doesn't equal the mean salary of female applicant, i.e., \(\mu_m \neq \mu_f\).

Compute Test Statistics

We're conducting a two sample t-test because we're comparing the means from two different groups. The test statistic for two-sample t-tests can be calculated by the formula: \[t = \frac{(\bar{x}_m - \bar{x}_f) - (\mu_m - \mu_f)}{\sqrt{(\frac{s^2_m}{n_m} + \frac{s^2_f}{n_f})}}\] where \(\bar{x}_m\) and \(\bar{x}_f\) are sample means, \(s_m\) and \(s_f\) are standard deviations, \(n_m\) and \(n_f\) are sample sizes. Given that, \(n_m = 63, n_f = 64, s_m = 5152, s_f = 7348, \bar{x}_m = 30238\) and \(\bar{x}_f = 26508\), we can substitute these values into the formula and calculate the value of t.

Find P-value

Using the calculated t value, look up the corresponding p-value in a t-distribution table. This p-value represents the probability that, if the null hypothesis is true, we would observe a sample as extreme as the one we have. If the p-value is less than our significance level (typically 0.05), we reject the null hypothesis in favor of the alternative.

Make Conclusion

If the p-value is less than 0.05, we reject the null hypothesis, which means there is a statistically significant difference between the mean salaries of male and female applicants, thus suggesting gender bias. However, if the p-value is greater than 0.05, we do not reject the null hypothesis, suggesting that there is not enough evidence to suggest gender bias. Either way, the conclusion will provide the answer to the question.

Key concepts

Two-Sample T-Test

Understanding the two-sample t-test is crucial when comparing two distinct groups to determine if there's a significant difference between their means. This statistical test is employed when you have two random samples that may have different variance, such as the study on gender bias in hiring.

In our example, the two groups are male and female applicants and their recommended salaries, which represent the populations we're analyzing. The decision to use a two-sample t-test is based on these being separate groups, where we can assume the individual observations within each group are independent of one another.

When employing the two-sample t-test, we use the following formula to compute the test statistic:
\[t = \frac{(\bar{x}_m - \bar{x}_f) - (\mu_m - \mu_f)}{\sqrt{(\frac{s^2_m}{n_m} + \frac{s^2_f}{n_f})}}\]Here, \(\bar{x}_m\) and \(\bar{x}_f\) refer to the sample means, \(s_m\) and \(s_f\) to the standard deviations, and \(n_m\) and \(n_f\) to the sample sizes of male and female applicants, respectively.

P-Value Interpretation

Interpreting the p-value is essential in understanding the results of our t-test and what they suggest about the presence of gender bias in hiring. The p-value answers the question: If the null hypothesis were true, what is the probability of observing a result as extreme as the one in our study?

A lower p-value indicates a rarer occurrence, suggesting that our observed data are unlikely under the null hypothesis. In a standard setting, if the p-value is below the chosen significance level (usually 0.05), we reject the null hypothesis. This rejection implies that there is sufficient evidence to support the alternative hypothesis, which indicates the presence of an effect or difference.

For instance, a very small p-value in the gender bias study points towards rejecting the null hypothesis of no difference in recommended salaries, thereby suggesting that gender bias is a plausible explanation for the observed difference in salaries. On the flip side, a p-value higher than 0.05 implies that there isn't enough evidence to reject the null hypothesis, and thus, we can't confidently state there is a bias based on the data presented.

Null and Alternative Hypotheses

Crafting the null and alternative hypotheses is a fundamental step in the hypothesis testing process. It lays the framework for what you are testing and what evidence you'll need to draw a conclusion about the population based on the sample data.

The null hypothesis (\(H_0\)) is the assumption that there is no effect, or no difference. It's essentially the status quo or the claim you're trying to challenge. In the context of gender bias in hiring, the null hypothesis posits that the mean salaries recommended for male and female applicants are equal; symbolically, we state \(\mu_m = \mu_f\).

The alternative hypothesis (\(H_a\) or \(H_1\)), on the other hand, represents the opposite claim. It is what you would conclude if you find sufficient evidence to reject the null hypothesis. In our gender bias example, the alternative hypothesis suggests that there is a difference in the mean salaries, which is indicated by \(\mu_m eq \mu_f\), and implies the potential existence of gender bias.

By setting up these hypotheses, we create a binary decision-making process where the collection and analysis of data lead us either to reject the null hypothesis, potentially accepting the alternative, or fail to reject it, suggesting that the alternative isn't supported by the sample evidence.