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.

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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​ satisifed; 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.

Answer 1

The sample was randomly and independently selected.   Successes in each group were 33 and 115 which are greater than 5. Same is true for the number of failures.

Hence, the right option is

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.

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Conducting the hypothesis test as below :

	n	Proportion (p)
Sample 1	n1 = 201229	p1 = 0.0002
Sample 2	n2 = 200745	p2 = 0.0006

The null and alternative hypotheses are
Ho :  P1 = P2
Ha :  P1 < P2
where P1, P2 are the population proportions for the proportions who developed polio in the treatment and placebo groups respectively

Let α = 0.05          ...Level of significance
Using the given formulae, we get

Pooled Proportion

p̂ = 0.0004
Standard Error (SE)

SE = 0.0001

z-statistic

z-statistic = z = -6.7563

Z-critical
For α = 0.05 we find the Less Than z-critical using z tables or Excel function NORM.S.INV
z critical = NORM.S.INV(0.05)
z critical = -1.645

P-value
For z = -6.7563 we find the Less Than p-value using z tables or Excel function NORM.S.DIST
p-value = NORM.S.DIST(-6.7563, TRUE)
p-value = 0

Decision
0 < 0.05
that is p-value is less than alpha.
Hence we Reject Ho.

Conclusion
There exists enough statistical evidence at α = 0.05 to show that the rate of polio is less for children given the Salk​ vaccine.

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