Question

At a Bloomburg City Council meeting, a plan to fund more swim safety programs was presented. The reasoning behind the request was that less than 40% of children under the age of 5 could pass a swim test. If this is true, the council will agree to fund more programs for these kids. The council decides to take a 200-person volunteer sample of children under 5 years in Bloomburg City and conduct a significance test for H0: p = 0.40 and Ha: p < 0.40, where p is the proportion of these children that can pass a swim test. They will perform a significance test at a significance level of α = 0.05 for the hypotheses.

Part A: Describe a Type II error that could occur. What impact could this error have on the situation?

Part B: Out of the 200 children under 5 that volunteered to take a swimming test, 87 passed, resulting in a p-value of 0.8438. What can you conclude from this p-value given the data of the 200 children is sufficient to perform a significance test for the hypotheses?

Part C: What possible defect in the study can you find in Part B? Explain.

Answer 1

Part A:

Type II Error: Failure to reject a false null hypothesis

H0: p = 0.40 ( 40% of children under the age of 5 could pass a swim test.)

HA: p < 0.40 ( less than 40% of children under the age of 5 could pass a swim test. ) (claim)

Suppose in reality, less than 40% of children under the age of 5 could pass a swim test. But, Bloomburg City Council wrongly conclude that 40% of children under the age of 5 could pass a swim test. Type II Error is committed in this situation.

The impact that could this error have on the situation is that Bloomburg City Council will not agree to fund more programs for these kids, whereas in reality it is very uch needed.

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 less than 40% of children under the age of 5 could pass a swim test.

Part C:

The possible defect in the study we can find in Part B is the the survey has Voluntary Response Bias.

Explanation: Since the participants are not selected through Simple Random Sampling (SRS) but the sample members are self - selected volunteers the resulting sample tends to overrepresent individuals who have strong opinion: here passing the swimming test.