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}} identify the rejection region(s). select the correct choice below and fill in the answer box(es) to complete your choice. (round to two decimal places as needed.) \\(\bigcirc\\) a. the rejection region is \\(\square < z < \square\\). \\(\bigcirc\\) b. the rejection region is \\(z > \square\\). \\(\bigcirc\\) c. the rejection regions are \\(z < \square\\) and \\(z > \square\\). \\(\bigcirc\\) d. the rejection region is \\(z < \square\\).

Explanation:

Step1: Identify the test type and significance level

This is a left - tailed z - test for a population proportion since the researcher's claim is that the proportion \(p<0.83\). The significance level \(\alpha = 0.05\).

Step2: Find the critical z - value

For a left - tailed test with \(\alpha=0.05\), we look for the z - value such that the area to the left of it is \(0.05\). From the standard normal distribution table (or using a calculator), the critical z - value \(z_{\alpha}\) for a left - tailed test with \(\alpha = 0.05\) is approximately \(- 1.645\approx - 1.65\) (rounded to two decimal places). In a left - tailed test, the rejection region is \(z < z_{\alpha}\), that is \(z < - 1.65\). So the correct option is D, and the rejection region is \(z < - 1.65\).

Answer:

D. The rejection region is \(z < - 1.65\)