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 necessary, fill in the answer box(es) to complete your choice.

○ a. the rejection region is z > \square.
(round to two decimal places as needed.)
○ b. the rejection regions are z < \square and z > \square.
(round to two decimal places as needed.)
○ c. the rejection region is z < \square.
(round to two decimal places as needed.)

Explanation:

Step1: Check Normal Sampling Distribution

To use a normal sampling distribution for a proportion, we check \( np \geq 5 \) and \( n(1 - p) \geq 5 \). Here, \( p = 0.24 \), \( n = 110 \).
\( np = 110\times0.24 = 26.4 \geq 5 \)
\( n(1 - p)=110\times(1 - 0.24)=110\times0.76 = 83.6 \geq 5 \). So normal sampling distribution can be used.

Step2: Determine Hypotheses and Rejection Region

The claim is \( p \geq 0.24 \), so the null hypothesis \( H_0: p = 0.24 \), alternative hypothesis \( H_a: p < 0.24 \) (left - tailed test). For a left - tailed test with \( \alpha = 0.08 \), we find the z - score such that \( P(Z < z)=\alpha = 0.08 \). Using the standard normal table or calculator, \( z_{\alpha}=z_{0.08}\approx - 1.41 \) (since \( P(Z < - 1.41)\approx0.08 \)). The rejection region is \( z < - 1.41 \), so option C is correct, and the value is - 1.41.

Answer:

C. The rejection region is \( z < \boldsymbol{-1.41}\)