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.
$z = \frac{x - np}{\sqrt{npq}}$

(b) find the critical value(s) and identify the rejection region(s).
identify the critical value(s) for this test.
$z_0 = \square$
(round to two decimal places as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Determine Test Type

This is a left - tailed z - test (since the claim is \(p<0.83\)). For a left - tailed test with \(\alpha = 0.05\), we need to find the z - value such that the area to the left of it is \(0.05\).

Step2: Use Z - Table or Calculator

We can use the standard normal distribution table (z - table) or a calculator with a normal distribution function. The z - value corresponding to a left - tailed area of \(0.05\) is found by looking for the value of \(z\) where \(P(Z < z_0)=0.05\). From the standard normal table, the critical value \(z_0=- 1.645\) (rounded to two decimal places, it can also be approximated as - 1.64 or - 1.65, but the more accurate value for \(\alpha = 0.05\) left - tailed is - 1.645).

Answer:

\(z_0=-1.645\) (or - 1.64, - 1.65 depending on rounding conventions, but - 1.645 is the precise value for \(\alpha = 0.05\) left - tailed test)