Question

for the following information, determine whether a normal sampling distribution can be used, where p is the population proportion, α is the level of significance, \hat{p} is the sample proportion, and n is the sample size. if it can be used, test the claim.

claim: p ≥ 0.24; α = 0.08. sample statistics: \hat{p} = 0.20, n = 110

if a normal sampling distribution can be used, identify the critical value(s) for this test. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. z₀ = \square
(round to two decimal places as needed. use a comma to separate answers as needed.)

b. a normal sampling distribution cannot be used.

Explanation:

Step1: Check Normal Sampling Condition

To use a normal sampling distribution for a proportion, we need \( np \geq 5 \) and \( n(1 - p) \geq 5 \). Here, \( p = 0.24 \), \( n = 110 \).
\( np = 110 \times 0.24 = 26.4 \geq 5 \)
\( n(1 - p)=110\times(1 - 0.24)=110\times0.76 = 83.6 \geq 5 \). So normal sampling is valid.

Step2: Determine Test Type and Critical Value

The claim is \( p \geq 0.24 \), so it's a left - tailed test (since the alternative hypothesis \( H_a:p < 0.24 \)). The significance level \( \alpha=0.08 \).
For a left - tailed test, the critical value \( z_0 \) is the z - score such that \( P(Z < z_0)=\alpha = 0.08 \).
Using the standard normal table or a calculator, \( z_{0.08}\approx - 1.41 \) (because \( P(Z < - 1.41)\approx0.08 \)).

Answer:

\( z_0=-1.41 \)