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Chapter 9: Problem 32

Researchers sent 5000 resumes in response to job ads that appeared in the Boston Globe and Chicago Tribune. The resumes were identical except that 2500 of them had "white sounding" first names, such as Brett and Emily, whereas the other 2500 had "black sounding" names such as Tamika and Rasheed. The resumes of the first type elicited 250 responses and the resumes of the second type only 167 responses (these numbers are very consistent with information that appeared in a Jan. 15, 2003, report by the Associated Press). Does this data strongly suggest that a resume with a "black" name is less likely to result in a response than is a resume with a "white" name?

Short Answer

Expert verified
Yes, the data suggests that resumes with "black" names receive fewer responses, supporting the alternative hypothesis.

Step by step solution

01

Define the Hypotheses

The first step is to establish the null and alternative hypotheses. The null hypothesis ( H_0) is that the names on the resumes have no effect on response rates, i.e., both groups have the same response rate. The alternative hypothesis ( H_a) is that the response rates are different between the two groups, particularly that 'black sounding' names result in fewer responses than 'white sounding' names.
02

Determine Response Rates

Calculate the response rate for each group. For 'white sounding' names, the response rate is \(\frac{250}{2500} = 0.10\) or 10%. For 'black sounding' names, it is \(\frac{167}{2500} \approx 0.0668\) or 6.68%.
03

Set Up the Test Statistic

To examine if the difference in response rates is significant, use a two-proportion z-test. The test statistic (z) is calculated using the formula: \[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \]where \(\hat{p}_1 = 0.10\), \(\hat{p}_2 = 0.0668\), \(\hat{p} = \frac{250 + 167}{2500 + 2500}\) is the combined proportion, and \(n_1 = n_2 = 2500\).
04

Calculate the Pooled Proportion

Calculate the pooled proportion \(\hat{p}\) which combines the data from both groups: \[ \hat{p} = \frac{250 + 167}{5000} = 0.0834\]
05

Compute the Standard Error

Calculate the standard error part of the formula using the pooled proportion: \[ SE = \sqrt{0.0834 \times (1 - 0.0834) \times \left( \frac{1}{2500} + \frac{1}{2500} \right)} = \sqrt{0.0834 \times 0.9166 \times 0.0008}\]
06

Calculate the Z-score

Compute the z-score using the response rates and standard error: \[ z = \frac{0.10 - 0.0668}{SE} \]First, calculate the standard error from the previous step, then substitute it to find the z-value.
07

Determine the P-value and Conclusion

Using the z-score from Step 6, determine the p-value associated with this z-score. If the p-value is less than the significance level (commonly 0.05), reject the null hypothesis. This would indicate that the difference in response rates is statistically significant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Two-Proportion Z-Test
When comparing two independent proportions, such as the response rates to resumes with different names, the two-proportion z-test is an ideal statistical tool. This type of test helps determine if there's a significant difference between the two proportions.

The test involves calculations based on the difference between the sample proportions. In this case, we compare the response rates for resumes with 'white sounding' names and 'black sounding' names.
  • The formula for the test statistic (z) is:\[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]where \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions of the two groups.
  • To perform the test, we need to calculate the pooled proportion and standard error.
Once computed, these values will help us determine if the observed difference is statistically significant.
Response Rates
In the context of statistical analysis, response rates are key indicators of interest. They expose the proportion of positive outcomes in each group. In our scenario, two groups are being assessed:

  • "White sounding" names resulted in 250 responses from 2500 resumes, giving a response rate of \( \frac{250}{2500} = 0.10 \) or 10%.
  • "Black sounding" names resulted in 167 responses from 2500 resumes, with a response rate of \( \frac{167}{2500} \approx 0.0668 \) or 6.68%.
Knowing the response rates is crucial for calculating the difference between the two groups. This difference informs us whether the outcome for one group is likely due to chance or a significant factor.
Null and Alternative Hypotheses
When examining differences in statistical data, defining the null and alternative hypotheses is an important initial step. These hypotheses form the foundation of hypothesis testing.

  • The **Null Hypothesis (\( H_0 \))** assumes that there is no effect or difference between groups. In this study, the null hypothesis is that the response rates are the same for both 'white sounding' and 'black sounding' names.
  • The **Alternative Hypothesis (\( H_a \))** proposes that there is a significant effect. Here, it suggests 'black sounding' names get fewer responses than 'white sounding' names.
Testing these hypotheses aids in determining whether the observed differences are statistically backed or if they arose by mere chance.
Pooled Proportion
The pooled proportion is a weighted average of the sample proportions from the two groups. It provides a common estimate of the proportion under the assumption that the null hypothesis is true.

To calculate the pooled proportion (\( \hat{p} \)), aggregate the successes from both groups and divide by the total number of observations.
  • In the scenario presented, the formula is:\[ \hat{p} = \frac{250 + 167}{5000} = 0.0834 \]
  • This pooled proportion is then used to find the standard error, which plays a crucial role in the calculation of the z-score.
The pooled proportion helps contextualize the expected variation between groups assuming the null hypothesis holds true.

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Most popular questions from this chapter

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