Question

(06.01, 06.02 MC) A recent report states that less than 25% of the adults in Clover City were able to pass a sign language test. Consequently, the city's communications department is trying to convince the city's education panel to fund more sign language programs. The panel will fund more programs only if the communications department can provide convincing evidence that the report is true.

Members of the Communications Department plan to collect data from a sample of 175 adult residents in Clover City. A test of significance will be conducted at a significance level of α = 0.05 for the hypotheses H0: p = 0.25 and Ha: p < 0.25, where p is the proportion of adult residents in the city who are able to pass the sign language test.

Part A: Describe a Type II error in the context of the study and the consequence of making this type of error. (3 points)

Part B: Members of the communications department recruit 175 adult residents who volunteer to take the sign language test. The test is passed by 54 of the 175 volunteers, resulting in a p-value of 0.96 for the given hypotheses. If it is reasonable to conduct a test of significance for the given hypotheses using the data collected from the 175 volunteers, what does the p-value of 0.96 lead you to conclude? (4 points)

Part C: Describe the primary flaw in the study described in part B and explain why it is a concern. (3 points) (10 points)

Answer 1

A.

Type II error is failure to reject the null hypothesis when the null hypothesis is false.

In the context of the study, type II error is that we conclude that there is no evidence that less than 25% of the adults in Clover City were able to pass a sign language test but in reality that proportion of the adults in Clover City who are able to pass a sign language test is less than 25%.

The consequence is that the panel will not fund more programs because the communications department cannot provide convincing evidence that the report is true. But in reality the report claim is true, and there is a need to fund more sign language programs.

B.

Since the 175 adult residents volunteered to take the sign language test and is not a random sample. It is not reasonable to conduct a test of significance for the given hypotheses using the data collected from the 175 volunteers.

C.

The primary flaw in the study is Voluntary response bias. Voluntary response bias occurs because sample members are self-selected volunteers, to take the sign language test. The volunteered sample may be expert in sign language test and tends to overestimate the proportion of adult residents in the city who are able to pass the sign language test. This will led to increase in probability of failure to reject the null hypothesis (increase in p-value) and consequently there would be increase in probability of Type II error.