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 > 0.64; α = 0.04. sample statistics: \hat{p} = 0.72, n = 275

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.

\bigcirc a. \\( z_0 = \square \\)
(round to two decimal places as needed. use a comma to separate answers as needed.)
\bigcirc b. a normal sampling distribution cannot be used.

if a normal sampling distribution can be used, identify the rejection region(s). select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

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

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, \( p = 0.64 \), \( n = 275 \).
Calculate \( np = 275\times0.64 = 176 \geq 5 \) and \( n(1 - p)=275\times(1 - 0.64)=275\times0.36 = 99 \geq 5 \). So normal sampling is valid.

Step2: Find Critical Value (Right - Tailed Test)

The test is right - tailed with \( \alpha = 0.04 \). The critical value \( z_0 \) is the z - score such that \( P(Z > z_0)=\alpha = 0.04 \), so \( P(Z\leq z_0)=1 - 0.04 = 0.96 \).
Using the standard normal table or calculator, \( z_0\approx1.75 \) (since \( \Phi(1.75)\approx0.9599\approx0.96 \)).

Step3: Determine Rejection Region

For a right - tailed test with critical value \( z_0 = 1.75 \), the rejection region is \( z>z_0 \), i.e., \( z > 1.75 \).

Answer:

Critical Value:

A. \( z_0 = 1.75 \)

Rejection Region:

A. The rejection region is \( z > 1.75 \)