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 ≥ 0.35; α = 0.06. sample statistics: \hat{p} = 0.30, n = 120

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. z_0 = \square
(round to two decimal places as needed. use a comma to separate answers as needed.)
○ 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.

○ a. the rejection region is z > \square.
(round to two decimal places as needed.)
○ b. partially visible, but the main problem is about normal sampling distribution for proportion test

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, \( p = 0.35 \), \( n = 120 \).
Calculate \( np = 120\times0.35 = 42 \geq 5 \) and \( n(1 - p)=120\times(1 - 0.35)=120\times0.65 = 78 \geq 5 \). So normal sampling is valid.

Step2: Determine Test Type and Critical Value

The claim is \( p \geq 0.35 \), so it's a left - tailed test? Wait, no: the alternative hypothesis for \( H_0:p = 0.35 \), \( H_a:p\lt0.35 \) (since claim is \( p\geq0.35 \), if we reject \( H_0 \) when \( p\lt0.35 \)). The significance level \( \alpha = 0.06 \). For a left - tailed test, the critical value \( z_0 \) is such that \( P(Z\lt z_0)=\alpha = 0.06 \). Using the standard normal table, \( z_0=\text{invNorm}(0.06)\approx - 1.55 \)? Wait, no, wait: Wait, the claim is \( p\geq0.35 \), so the null hypothesis \( H_0:p = 0.35 \), alternative \( H_a:p\lt0.35 \) (left - tailed). But wait, let's re - check the critical value. The significance level \( \alpha = 0.06 \), for a left - tailed test, the critical value \( z_{\alpha} \) is the value where the area to the left is \( \alpha \). Looking up in the z - table, the z - score corresponding to an area of \( 0.06 \) to the left is approximately \( z=-1.55 \)? Wait, no, actually, for a left - tailed test with \( \alpha = 0.06 \), the critical value \( z_0 \) is \( z=-1.55 \) (using calculator: \( \text{invNorm}(0.06)\approx - 1.5548\approx - 1.55 \)). But wait, the first part: when we check the normal condition, we used \( p = 0.35 \), which is correct for the null hypothesis.

Step3: Rejection Region

For a left - tailed test with critical value \( z_0=-1.55 \), the rejection region is \( z\lt - 1.55 \). But wait, the options for critical value: option A is \( z_0=\) (we need to find it). Wait, maybe I made a mistake in the test direction. Wait, the claim is \( p\geq0.35 \), so the null hypothesis \( H_0:p = 0.35 \), alternative \( H_a:p\lt0.35 \) (left - tailed). The significance level \( \alpha = 0.06 \). So the critical value is \( z_{\alpha} \), where \( P(Z\lt z_{\alpha})=\alpha = 0.06 \). Using the standard normal distribution, \( z_{\alpha}=\text{invNorm}(0.06)\approx - 1.55 \).

Answer:

First, for the critical value:
A. \( z_0=-1.55 \) (rounded to two decimal places)

For the rejection region: The rejection region is \( z\lt - 1.55 \), so if the option for rejection region is like "The rejection region is \( z\lt - 1.55 \)", but based on the given options, for the critical value part, the answer for the critical value option (A) is \( z_0 = - 1.55 \), and a normal sampling distribution can be used (so option B is wrong).