Question

A large restaurant is being sued for age discrimination because 15% of newly hired candidates are between the ages of 30 years and 50 years when 50% of all applicants were in that age bracket. You plan to use hypothesis testing to determine whether there is significant evidence that the company's hiring practices are discriminatory. Part A: State the null and alternative hypotheses for the significance test. (2 points) Part B: In the context of the problem, what would a Type I error be? A Type II error? (2 points) Part C: If the hypothesis is tested at a 1% level of significance instead of 5%, how will this affect the power of the test? (3 points) Part D: If the hypothesis is tested based on the hiring of 1,000 employees rather than 100 employees, how will this affect the power of the test? (3 points)

Answer 1

Answer:

Part A:

The null and alternative hypothesis are:

$H_0: \pi=0.5\\\\H_1: \pi\neq 0.5$

Part B:

- A Type I error is when the null hypothesis is rejected although it is true. In this case, it would mean that we conclude that the hiring process is discriminatory, when in reality it is a random result and the process is not discriminatory.

- A Type II error is when the null hypothesis fails to be rejected although it is false. In this case, the hiring process is discriminatory, but statistically the result is not significant enough to prove that.

Part C:

A reduction in the significance level causes a reduction in the power of the test.

Part D:

The power of the test is increased with a larger sample.

Step-by-step explanation:

We have a restaurant with hire a proprotion of 15 % of people in the age ragne of 30-50 years. The expected proportion, according to the applicants, is 50%.

The test will tell us if the actual 15% is a result of a discriminatory practice or a random result.

Part A:

The null and alternative hypothesis are:

$H_0: \pi=0.5\\\\H_1: \pi\neq 0.5$

Part B:

- A Type I error is when the null hypothesis is rejected although it is true. In this case, it would mean that we conclude that the hiring process is discriminatory, when in reality it is a random result and the process is not discriminatory.

- A Type II error is when the null hypothesis fails to be rejected although it is false. In this case, the hiring process is discriminatory, but statistically the result is not significant enough to prove that.

Part C:

The power of an hypothesis test is the probability that the test rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true.

If the significance level is reduced (from 5% to 1%), the rejection region is reduced, so the probability of rejecting the null hypothesis is also reduced.

Then, a reduction in the significance level causes a reduction in the power of the test.

Part D:

A bigger sample size gives robustness to the sample statistic. Then, if the alternative hypothesis is true, the probabilities of detecting the effect are increased with increased sample size.

Then, the power of the test is increased with a larger sample.

Answer 2

The test will tell us if the actual 15% is a result of a discriminatory practice or a random result.

Part A: $H_0= \mu=0.5\\H_0= \mu\neq 0.5$

Part B: A Type I error is when the null hypothesis is rejected although it is true. A Type II error is when the null hypothesis fails to be rejected although it is false.

Part C: A reduction in the significance level causes a reduction in the power of the test.

Part D: The power of the test is increased with a larger sample.

We have a restaurant with hire a proprotion of 15 % of people in the age ragne of 30-50 years. The expected proportion, according to the applicants, is 50%.

The test will tell us if the actual 15% is a result of a discriminatory practice or a random result.

Part A: The null and alternative hypothesis are:

$H_0= \mu=0.5\\H_0= \mu\neq 0.5$

Part B:

- A Type I error is when the null hypothesis is rejected although it is true. In this case, it would mean that we conclude that the hiring process is discriminatory, when in reality it is a random result and the process is not discriminatory.

- A Type II error is when the null hypothesis fails to be rejected although it is false. In this case, the hiring process is discriminatory, but statistically the result is not significant enough to prove that.

Part C: The power of an hypothesis test is the probability that the test rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true. If the significance level is reduced (from 5% to 1%), the rejection region is reduced, so the probability of rejecting the null hypothesis is also reduced. Then, a reduction in the significance level causes a reduction in the power of the test.

Part D: A bigger sample size gives robustness to the sample statistic. Then, if the alternative hypothesis is true, the probabilities of detecting the effect are increased with increased sample size. Then, the power of the test is increased with a larger sample.

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