Question

in the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. the treatment group consisted of 201,229 children given the salk vaccine for polio, and the other 200,745 children were given a placebo. among those in the treatment group, 33 developed polio, and among those in the placebo group, 115 developed polio. if we want to use the methods for testing a claim about two population proportions to test the claim that the rate of polio is less for children given the salk vaccine, are the requirements for a hypothesis test satisfied? explain. choose the correct answer below. a. the requirements are not satisfied; the difference between the rates of those that developed polio in the two groups is not statistically significant b. the requirements are satisfied; the samples are simple random samples that are independent, and for each of the two groups, the number of successes is at least 5 and the number of failures is at least 5 c. the requirements are satisfied; the samples are simple random samples that are independent, and for each of the two groups, the sample size is at least 1000 d. the requirements are not satisfied; the samples are not simple random samples that are independent

Explanation:

To determine if the requirements for a hypothesis test about two population proportions are satisfied:

1. Simple Random Samples (SRS) & Independence: The children were randomly assigned to two groups, so samples are independent SRS.
2. Success/Failure Condition: For the treatment group (n₁ = 201,229, x₁ = 33):

• Successes (x₁ = 33) ≥ 5, Failures (n₁ - x₁ = 201,229 - 33 = 201,196) ≥ 5.

For the placebo group (n₂ = 200,745, x₂ = 115):

• Successes (x₂ = 115) ≥ 5, Failures (n₂ - x₂ = 200,745 - 115 = 200,630) ≥ 5.

Thus, the “at least 5 successes and failures” requirement is met.

Option B correctly states the requirements are satisfied: samples are independent SRS, and each group has ≥5 successes and failures.

Answer:

B. The requirements are satisfied; the samples are simple random samples that are independent, and for each of the two groups, the number of successes is at least 5 and the number of failures is at least 5.