Question

a medical researcher says that less than 72% of adults in a certain country think that healthy children should be required to be vaccinated. in a random sample of 500 adults in that country, 69% think that healthy children should be required to be vaccinated. at α = 0.05, is there enough evidence to support the researchers claim? complete parts (a) through (d) below.

(round to two decimal places as needed. use a comma to separate answers as needed.)
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.)
a. the rejection region is z <

b. the rejection regions are z < and z >

c. the rejection region is z >

d. the rejection region is < z <

Explanation:

Step1: Identify the test type

This is a left - tailed hypothesis test because the researcher's claim is that the proportion \(p<0.72\). The null hypothesis \(H_0:p = 0.72\) and the alternative hypothesis \(H_a:p<0.72\).

Step2: Determine the significance level and critical value

For a left - tailed test with \(\alpha = 0.05\), we need to find the \(z\) - score such that the area to the left of the \(z\) - score is \(0.05\). Looking up in the standard normal distribution table (or using a calculator with a normal - distribution function), the critical value \(z_{\alpha}\) for a left - tailed test with \(\alpha=0.05\) is approximately \(- 1.645\) (rounded to two decimal places, it is \(-1.65\)). The rejection region for a left - tailed test is \(z < z_{\alpha}\), so the rejection region is \(z<- 1.65\). And the correct choice is A, with the value in the box being \(-1.65\).

Answer:

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