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 \\(\alpha = 0.05\\), is there enough evidence to support the researcher’s claim? complete parts (a) through (d) below.

the rejection region is \\(\square < z < \square\\).

(c) find the standardized test statistic \\(z\\).

\\(z = -1.49\\) (round to two decimal places as needed.)

(d) decide whether to reject or fail to reject the null hypothesis and interpret the decision in the context of the original claim.

\\(\square\\) the null hypothesis. there \\(\square\\) enough evidence to \\(\square\\) the researcher’s claim.

Explanation:

Step1: Identify Hypotheses

The claim is \( p < 0.72 \), so null \( H_0: p = 0.72 \), alternative \( H_a: p < 0.72 \) (left - tailed test).

Step2: Find Critical Value

For \( \alpha = 0.05 \) (left - tailed), critical \( z_{\alpha}=- 1.645 \). Rejection region: \( z < - 1.645 \).

Step3: Compare Test Statistic and Critical Value

Test statistic \( z=-1.49 \). Since \( - 1.49>-1.645 \), we fail to reject \( H_0 \).

Step4: Interpret the Decision

Failing to reject \( H_0 \) means there is not enough evidence to support the researcher's claim that less than 72% of adults think healthy children should be vaccinated.

Answer:

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