Question

A recent report states that less than 40% of the children younger than 5 years in Bloomburg City were able to pass a swimming test. Consequently, the city's aquatics department is trying to convince the city council to fund more swimming programs. The council will fund more programs only if the aquatics department can provide convincing evidence that the report is true.

Members of the aquatics department plan to collect data from a sample of 200 children younger than 5 years who live in Bloomburg City. A test of significance will be conducted at a significance level of α = 0.05 for the hypotheses H₀: p = 0.40 and H_a: p < 0.40, where p is the proportion of children who live in the city and are able to pass the swimming test.

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

Part B: Members of the aquatics department recruit 200 residents younger than 5 years who volunteer to take the swimming test. The test is passed by 87 of the 200 volunteers, resulting in a p-value of 0.8438 for the given hypotheses. If it is reasonable to conduct a test of significance for the given hypotheses using the data collected from the 200 volunteers, what does the p-value of 0.8438 lead you to conclude?

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

Answer 1

Part A:

H0: Null Hypothesis: p = 0.40

HA: Alternative Hypothesis: p < 0.40

Type II Error: Failure to reject false null hypothesis.

Suppose in reality the proportion of children who live in the city and are able to pass the swimming test is less than 40%. But,the aquatics department wrongly concludes that the proportion of children who live in the city and are able to pass the swimming test is 40%. Type II Error is committed in this situation.

The consequence of making this type of error is that the city's aquatic department is unable to convince the city council and they will not get funds, which is in fact needed in reality.

Part B:
Since p - value = 0.8438 is greater than = 0.05, the difference is not significant, Fail to reject null hypothesis.

Conclusion:

The data do not support the claim that the proportion of children who live in the city and are able to pass the swimming test is less than 40%.

Part C:

The primary flaw in the study is that the inherent talent of a child in swimming is not the same for all children with or without swimming programs. For reliable conclusion we should choose children of equal inherent talent in swimming for the experimentation.