Question

a medical researcher says that less than 83% of adults in a certain country think that healthy children should be required to be vaccinated. in a random sample of 400 adults in that country, 80% think that healthy children should be required to be vaccinated. at α = 0.05, is there enough evidence to support the researcher’s claim? complete parts (a) through (d) below. when you know the number of successes x, the sample size n, and the population proportion p, it can be easier to use the formula shown below to find the standardized test statistic when using a z - test for a population proportion p. $z = \frac{x - np}{\sqrt{npq}}$ (d) decide whether to reject or fail to reject the null hypothesis and interpret the decision in the context of the original claim. dropdown the null hypothesis. there dropdown enough evidence to dropdown the researcher’s claim. when performing this test using the formula $z = \frac{\hat{p} - p}{\sqrt{\frac{pq}{n}}}$, the test statistic is z = - 1.60 and the decision is to fail to reject the null hypothesis. are these results the same?

Explanation:

Step1: Recall Hypothesis Test Decision

For a left - tailed test (since the claim is \(p < 0.83\)) with \(\alpha=0.05\), the critical value \(z_{\alpha}=- 1.645\). The test statistic \(z=-1.60\) (from the problem). Since \(-1.60>- 1.645\) (the test statistic is not in the rejection region), we fail to reject the null hypothesis.

Step2: Interpret the Decision

Failing to reject the null hypothesis means that there is not enough evidence to support the alternative hypothesis (which is the researcher's claim that \(p < 0.83\)).

Answer:

Fail to reject the null hypothesis. There is not enough evidence to support the researcher’s claim.

For the second part about whether the results are the same: The two formulas for the z - test of proportion are equivalent. \(\hat{p}=\frac{x}{n}\), so \(x = n\hat{p}\). Substituting \(x = n\hat{p}\) into \(z=\frac{x - np}{\sqrt{npq}}\) gives \(z=\frac{n\hat{p}-np}{\sqrt{npq}}=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}\), which is the other formula. So the test statistic values should be the same (up to rounding) and the decision (fail to reject) is the same. So the test statistic is the same, and the decision is the same.