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 standardized test statistic z. select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\bigcirc a. z = \square
(round to two decimal places as needed.)
\bigcirc b. a normal sampling distribution cannot be used.

if a normal sampling distribution can be used, decide whether to reject or fail to reject the null hypothesis and interpret the decision. choose the correct answer below.

\bigcirc a. reject the null hypothesis. there is enough evidence to support the claim.
\bigcirc b. fail to reject the null hypothesis. there is not enough evidence to support the claim.
\bigcirc c. fail to reject the null hypothesis. there is enough evidence to support the claim.

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, \( p = 0.64 \), \( n = 275 \).
\( np = 275\times0.64 = 176 \geq 5 \)
\( n(1 - p)=275\times(1 - 0.64)=275\times0.36 = 99 \geq 5 \).
So, normal sampling distribution can be used.

Step2: State Hypotheses

Null hypothesis: \( H_0: p \leq 0.64 \)
Alternative hypothesis: \( H_a: p > 0.64 \) (claim, right - tailed test)

Step3: Calculate Standard Error (SE)

The formula for the standard error of the sample proportion is \( SE=\sqrt{\frac{p(1 - p)}{n}} \)
Substitute \( p = 0.64 \), \( n = 275 \):
\( SE=\sqrt{\frac{0.64\times(1 - 0.64)}{275}}=\sqrt{\frac{0.64\times0.36}{275}}=\sqrt{\frac{0.2304}{275}}\approx\sqrt{0.0008378}\approx0.0289 \)

Step4: Calculate z - statistic

The formula for the z - statistic for a sample proportion is \( z=\frac{\hat{p}-p}{SE} \)
Substitute \( \hat{p}=0.72 \), \( p = 0.64 \), \( SE\approx0.0289 \):
\( z=\frac{0.72 - 0.64}{0.0289}=\frac{0.08}{0.0289}\approx2.77 \) (rounded to two decimal places)

Step5: Find Critical Value

For a right - tailed test with \( \alpha = 0.04 \), the critical value \( z_{\alpha} \) is the z - score such that \( P(Z > z_{\alpha})=0.04 \). So, \( P(Z\leq z_{\alpha}) = 1 - 0.04=0.96 \). Looking up in the z - table, \( z_{\alpha}\approx1.75 \) (since \( P(Z\leq1.75)\approx0.9599\approx0.96 \))

Step6: Make a Decision

Since the calculated z - statistic (\( z\approx2.77 \)) is greater than the critical value (\( z_{\alpha}\approx1.75 \)), we reject the null hypothesis. There is enough evidence to support the claim that \( p > 0.64 \).

Answer:

For the z - statistic: A. \( z = 2.77 \)
For the hypothesis test decision: A. Reject the null hypothesis. There is enough evidence to support the claim.