Question

an education researcher claims that 60% of college students work year-round. in a random sample of 500 college students, 300 say they work year-round. at α = 0.01, is there enough evidence to reject the researcher’s 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.
z₀ = \square
(round to two decimal places as needed. use a comma to separate answers as needed )

Explanation:

Step1: Determine Test Type

This is a two - tailed z - test for a proportion (since \(H_0:p = 0.60\) and \(H_a:p
eq0.60\)) with \(\alpha=0.01\). For a two - tailed test, we split \(\alpha\) equally between the two tails. So \(\alpha/2=0.01/2 = 0.005\).

Step2: Find Critical Values

We need to find the \(z\) - scores such that the area in each tail is \(0.005\). The area to the left of the lower critical value is \(0.005\), and the area to the left of the upper critical value is \(1 - 0.005=0.995\).
Using the standard normal distribution table (or \(z\) - table), the \(z\) - score corresponding to an area of \(0.005\) in the left tail is \(z=- 2.58\) (because \(P(Z < - 2.58)\approx0.005\)) and the \(z\) - score corresponding to an area of \(0.995\) in the left tail is \(z = 2.58\) (because \(P(Z<2.58)\approx0.995\)).

Answer:

\(z_{0}=-2.58,2.58\)