Question

consider the following hypotheses
$h_{0}:p = 0.44$
$h_{1}:p
eq0.44$
given that $hat{p}=0.39$, $n = 500$, and $alpha=0.10$, answer the following questions
a. determine the critical value(s) and the test statistic. what conclusion should be drawn?
b. determine the p - value for this test.
(note: $x=hat{p}n$)
a. determine the critical value(s) or the test statistic.
$z_{\frac{alpha}{2}}=square$
(use a comma to separate answers as needed. round to three decimal places as needed.)
calculate the test statistic.
$z_{0}=square$ (round to two decimal places as needed.)
choose the correct answer below.
a. do not reject $h_{0}$. there is insufficient evidence that $p
eq0.44$.
b. do not reject $h_{0}$. there is sufficient evidence that $p
eq0.44$.
c. reject $h_{0}$. there is sufficient evidence that $p
eq0.44$.

Explanation:

Step1: Find critical values

For a two - tailed test with \(\alpha = 0.10\), the significance level in each tail is \(\frac{\alpha}{2}=0.05\).
Using the standard normal distribution \(Z\), the critical values \(z_{\frac{\alpha}{2}}\) are \(z_{0.05}\).
From the standard normal table, \(z_{0.05}=\pm1.645\).

Step2: Calculate the test statistic

The formula for the test statistic \(z_0\) in a proportion test is \(z_0=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Given \(\hat{p}=0.39\), \(p = 0.44\), \(n = 500\)
First, calculate the denominator \(\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.44\times(1 - 0.44)}{500}}=\sqrt{\frac{0.44\times0.56}{500}}=\sqrt{\frac{0.2464}{500}}\approx\sqrt{0.0004928}\approx0.0222\)
Then \(z_0=\frac{0.39 - 0.44}{0.0222}=\frac{- 0.05}{0.0222}\approx - 2.25\)

Step3: Make a decision

Since \(|z_0|=2.25>1.645\) (the critical value), we reject the null hypothesis \(H_0\)

Answer:

\(z_{\frac{\alpha}{2}}=-1.645,1.645\)
\(z_0=-2.25\)
C. Reject \(H_0\). There is sufficient evidence that \(p
eq0.44\)