Question

consider testing a population proportion with these hypotheses:
$h_0:p = 0.40$
$h_a:p\
eq0.40$
a random sample of size 600 produces a sample proportion $\hat{p}=0.375$.
a. compute the test statistic and associated $p$-value. (round each value as instructed)
$z_0 = \square$ (round your answer to two decimal places)
$p$-value $=\square$ (round your answer to four decimal places)
b. at significance level $\alpha = 0.05$, choose the correct conclusion.
o a. do not reject $h_0$. there is insufficient evidence to conclude that $p\
eq0.40$.
o b. reject $h_0$. there is sufficient evidence to conclude that $p\
eq0.40$.
o c. reject $h_0$. there is insufficient evidence to conclude that $p\
eq0.40$.
o d. do not reject $h_0$. there is sufficient evidence to conclude that $p\
eq0.40$.
c. determine the rejection region using the critical value. (round your answers to two decimal places. enter none for the unused tail if the test is one - tailed.)
$z_0\leq\square$
$z_0\geq\square$
what can you conclude?
o a. do not reject $h_0$. there is sufficient evidence to conclude that $p\
eq0.40$.
o b. reject $h_0$. there is insufficient evidence to conclude that $p\
eq0.40$.
o c. reject $h_0$. there is sufficient evidence to conclude that $p\
eq0.40$.
. o d. do not reject $h_0$. there is insufficient evidence to conclude that $p\
eq0.40$.

Explanation:

Step1: Calculate the test statistic \( z_0 \)

The formula for the test statistic in a proportion test is \( z_0=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\).
Given \( n = 600\), \(\hat{p}=0.375\), \(p_0 = 0.40\).
First, calculate \(\sqrt{\frac{p_0(1 - p_0)}{n}}=\sqrt{\frac{0.40\times(1 - 0.40)}{600}}=\sqrt{\frac{0.40\times0.60}{600}}=\sqrt{\frac{0.24}{600}}=\sqrt{0.0004}=0.02\).
Then \(z_0=\frac{0.375 - 0.40}{0.02}=\frac{- 0.025}{0.02}=-1.25\).

Step2: Calculate the P - value

Since this is a two - tailed test (\(H_0:p = 0.40\) vs \(H_a:p
eq0.40\)), the P - value is \(P = 2\times P(Z\lt|z_0|)\).
Using the standard normal distribution table, \(P(Z\lt1.25)=0.1056\).
So \(P - value=2\times0.1056 = 0.2112\).

Step3: Determine the rejection region

For a two - tailed test with \(\alpha=0.05\), the critical values are \(z_{\alpha/2}=\pm1.96\). The rejection region is \(z_0\geq1.96\) or \(z_0\leq - 1.96\).

Step4: Make a conclusion

Since \(|z_0|=1.25\lt1.96\) and \(P - value = 0.2112\gt0.05\), we do not reject \(H_0\). There is insufficient evidence to conclude that \(p
eq0.40\).

Answer:

a. \(z_0=-1.25\), \(P - value = 0.2112\)
b. Do not reject \(H_0\). There is insufficient evidence to conclude that \(p
eq0.40\)
c. Rejection region: \(z_0\geq1.96\) or \(z_0\leq - 1.96\)