Question

The article "A 'White' Name Found to Help in Job Search" (Associated Press, January 15,2003 ) described an experiment to investigate if it helps to have a "whitesounding" first name when looking for a job. Researchers sent resumes in response to 5,000 ads that appeared in the Boston Globe and Chicago Tribune. The resumes were identical except that 2,500 of them used "white-sounding" first names, such as Brett and Emily, whereas the other 2,500 used "black- sounding" names such as Tamika and Rasheed. The 5,000 job ads were assigned at random to either the white-sounding name group or the blacksounding name group. Resumes with white-sounding names received 250 responses while resumes with black sounding names received only 167 responses. Do these data support the claim that the proportion receiving a response is significantly higher for resumes with "white-sounding" first names? (Hint: See Example 14.2 )

Short answer

The answer depends on the calculated z value. If it's greater than 1.645, the conclusion will be that the response rate for resumes with 'white-sounding' names is significantly higher than for resumes with 'black-sounding' names.

Step-by-step solution

Formulate Hypotheses

The Null Hypothesis: The responses for both groups are equal or (\( p_1 = p_2 \)). The Alternative Hypothesis: The response rate for resumes with white sounding names is significantly higher than for those with black sounding names (\( p_1 > p_2 \)). \( p_1 \) denotes the proportion for white sounding names, while \( p_2 \) denotes the proportion for black sounding names.

Calculate Proportions and Difference

Let's calculate the proportions for both groups. For white sounding names, 250 responses out of 2500 resumes gave a response rate of \( p_1 = 250/2500 = 0.1 \). For black sounding names, 167 responses out of 2500 resumes gave a response rate of \( p_2 = 167/2500 = 0.0668 \). The difference between both proportions is \( p_1 - p_2 = 0.1 - 0.0668 = 0.0332 \).

Calculate Test Statistic

The z-test statistic formula is: \( z = (p_1 - p_2)/ \sqrt{p(1-p)(1/n_1 + 1/n_2)} \), where \( p \) is the pooled proportion calculated as \( (x_1+x_2)/(n_1 + n_2) \). Here, \( x_1 = 250, x_2 = 167, n_1 = n_2 = 2500 \). Calculate p and substitute all values to get the z value.

Compare Test Statistic to Critical Value

For a significance level of 0.05 (typical), the critical z value (one-tailed) is about 1.645. If the calculated z value is greater than this critical value, we reject the null hypothesis.

Conclude

If the z value is greater than the critical value, we reject our null hypothesis and conclude that the response rate for resumes with 'white-sounding' names is significantly higher. If not, we do not have enough evidence to support this claim.

Key concepts

Null and Alternative Hypotheses

When performing hypothesis testing in statistics, we begin by formulating two contrasting statements: the null hypothesis and the alternative hypothesis. The null hypothesis, typically denoted as \( H_0 \), represents a hypothesis of no effect or no difference. It is the statement being tested and usually proposes that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error. For example, if we were investigating the response rates to resumes with different-sounding names, the null hypothesis would claim that the proportion of responses \( p_1 \) for 'white-sounding' names is equal to the proportion \( p_2 \) for 'black-sounding' names, expressed as \( p_1 = p_2 \).

Conversely, the alternative hypothesis, denoted as \( H_a \) or \( H_1 \), is a statement that indicates the presence of an effect or difference. In our case, the alternative hypothesis states that there is a significant difference in response rates, positing that the response rate for 'white-sounding' names is higher than for 'black-sounding' names, expressed as \( p_1 > p_2 \). Determining whether to reject the null hypothesis or not is based on the data and appropriate statistical tests.

Z-Test Statistic

The z-test statistic is a numerical measurement that is used to evaluate the difference between an observed statistic and its hypothesized population parameter when the sample mean and population standard deviation are known. It is calculated using a standard formula, which in the case of comparing two proportions is given by: \[ z = (p_1 - p_2) / \sqrt{p(1-p)(1/n_1 + 1/n_2)} \]

In this formula, \( p_1 \) and \( p_2 \) are the sample proportions, while \( p \) is the pooled sample proportion representing a weighted average of the two sample proportions. The terms \( n_1 \) and \( n_2 \) denote the sample sizes for each group respectively. The z-test statistic provides a way of determining how many standard deviations an element is from the mean. A larger absolute value of the z-test statistic indicates a larger difference between the compared proportions.

Statistical Significance

Statistical significance is a determination of whether the observed results in a study are likely due to something other than mere random chance. This concept plays a central role in hypothesis testing. 'Significance level' is the threshold against which we measure the p-value of our test results and is denoted by \( \alpha \), commonly set at 0.05 or 5%. This means we would expect to see the observed outcome by chance only 5% of the time or less if the null hypothesis were true.

To assess statistical significance, we compare the calculated z-test statistic to a critical value, which corresponds to the cut-off for what we consider rare enough to be considered statistically significant under the null hypothesis. If the calculated statistic exceeds this critical value in the direction implied by the alternative hypothesis, we reject the null hypothesis, deeming the results statistically significant. This forms the basis for drawing conclusions in the context of hypothesis testing.

Proportions Comparison

Comparison of proportions is often employed when two distinct groups are observed, and we aim to ascertain whether there is a statistically significant difference in the proportion of individuals with certain characteristics between the two groups. For example, comparing the response rates to resumes with different-sounding names involves calculating the proportion of positive responses for both 'white-sounding' and 'black-sounding' names.

In the scenario provided, the calculated proportions were \( p_1 = 0.1 \) for 'white-sounding' names and \( p_2 = 0.0668 \) for 'black-sounding' names. To determine if the observed difference is statistically significant, we use hypothesis testing techniques, such as calculating the z-test statistic and comparing against the critical value. This approach allows researchers to make informed decisions about whether the observed differences in proportions are meaningful or could have arisen by random variation.