Question

decide whether the normal sampling distribution can be used. if it can be used, test the claim about the population proportion p at the given level of significance α using the given sample statistics.
claim: p ≠ 0.24; α = 0.01; sample statistics: \\( \hat { p } = 0.17 \\), n = 200
a. \\( h _ { 0 } : p = 0.24 \\)
\\( h _ { a } : p ≠ 0.24 \\)
b. \\( h _ { 0 } : p ≥ 0.24 \\)
\\( h _ { a } : p < 0.24 \\)
c. \\( h _ { 0 } : p ≤ 0.24 \\)
\\( h _ { a } : p > 0.24 \\)
d. the test cannot be performed.
determine the critical value(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the critical value(s) is/are □
(round to two decimal places as needed. use a comma to separate answers as needed.)
b. the test cannot be performed.

Explanation:

Step1: Check if normal sampling distribution can be used

For a normal sampling distribution of \( \hat{p} \) to be used, we need \( np\geq5 \) and \( n(1 - p)\geq5 \).
Given \( p = 0.24 \), \( n=200 \)
\( np=200\times0.24 = 48\geq5 \)
\( n(1 - p)=200\times(1 - 0.24)=200\times0.76 = 152\geq5 \)
So, the normal sampling distribution can be used.

Step2: Set up hypotheses

The claim is \( p
eq0.24 \). The null hypothesis \( H_{0}\) is the statement of equality, and the alternative hypothesis \( H_{a}\) is the claim. So \( H_{0}:p = 0.24\) and \( H_{a}:p
eq0.24\) (which is option A)

Step3: Find critical values

Since \( \alpha=0.01 \) and the test is two - tailed (because \( H_{a}:p
eq0.24 \)), we find the \( z\) - values that leave an area of \( \frac{\alpha}{2}=0.005 \) in each tail.
Using the standard normal distribution table or a calculator, the critical values are \( z=\pm z_{\alpha/2}\)
\( z_{0.005}\) corresponds to the \( z\) - value such that \( P(Z>z)=0.005 \) and \( P(Z < - z)=0.005 \)
From the standard normal table, \( z=\pm 2.58\)

Answer:

A. The critical values are \(- 2.58,2.58\)