Question

a medical researcher says that less than 80% of adults in a certain country think that healthy children should be required to be vaccinated. in a random sample of 600 adults in that country, 78% think that healthy children should be required to be vaccinated. at \\( \alpha = 0.01 \\), is there enough evidence to support the researchers claim? complete parts (a) through (d) below
(b) find the critical value(s) and identify the rejection region(s)
identify the critical value(s) for this test.
\\( z _ { 0 } = - 2.33 \\)
(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 )
(c) find the standardized test statistic z.
\\( z = \square \\)
(round to two decimal places as needed )

Explanation:

Step1: Calculate the value of \(q\)

We know that \(p = 0.8\) (from the null hypothesis \(H_0:p\geq0.8\)), so \(q=1 - p=1 - 0.8 = 0.2\)

Step2: Calculate the standard error \(\sigma_{\hat{p}}\)

The formula for the standard error of a proportion is \(\sigma_{\hat{p}}=\sqrt{\frac{pq}{n}}\). Substituting \(p = 0.8\), \(q = 0.2\) and \(n = 600\), we get \(\sigma_{\hat{p}}=\sqrt{\frac{0.8\times0.2}{600}}=\sqrt{\frac{0.16}{600}}\approx0.0163\)

Step3: Calculate the \(z\) - statistic

The formula for the \(z\) - statistic in a proportion test is \(z=\frac{\hat{p}-p}{\sigma_{\hat{p}}}\). Here, \(\hat{p}=0.78\), \(p = 0.8\) and \(\sigma_{\hat{p}}\approx0.0163\). Then \(z=\frac{0.78 - 0.8}{0.0163}=\frac{- 0.02}{0.0163}\approx - 1.23\)

Answer:

\(z\approx - 1.23\)