Question

consider the following hypotheses
$h_0: p = 0.23$
$h_1: p>0.23$
given that $hat{p}=0.3$, $n = 130$, and $\alpha=0.01$, answer the following questions
a. determine the critical value(s), the test statistic. what conclusion should be drawn?
b. determine the p - value for this test.
(note: $x=hat{p}n$)
calculate the test statistic.
$z_0 = 1.90$ (round to two decimal places as needed.)
what conclusion should be drawn?
a. do not reject $h_0$. there is insufficient evidence that $p>0.23$.
b. reject $h_0$. there is insufficient evidence that $p>0.23$.
c. do not reject $h_0$. there is sufficient evidence that $p>0.23$.
d. reject $h_0$. there is sufficient evidence that $p>0.23$.
b. p - value = (round to three decimal places as needed.)

Explanation:

Step1: Find the critical value

Since \(H_1:p > 0.23\) (right - tailed test) and \(\alpha=0.01\), from the standard normal distribution table \(z_{\alpha}=z_{0.01}\). The critical value \(z_{\alpha}\) is \(2.33\) (because the area to the right of \(z = 2.33\) is \(0.01\)).

Step2: Calculate the test statistic

The formula for the test statistic \(z_0\) in a one - sample proportion test is \(z_0=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}}\)
Given \(\hat{p}=0.3\), \(p = 0.23\), \(n = 130\)

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Step3: Draw a conclusion

Since \(z_0=1.90 0.23\)

Step4: Calculate the p - value

For a right - tailed test, \(p-\text{value}=P(Z>z_0)\)
Since \(z_0 = 1.90\), \(p-\text{value}=1 - P(Z\leq1.90)\)
From the standard normal table, \(P(Z\leq1.90)=0.9713\)
So \(p-\text{value}=1 - 0.9713=0.029\)

Answer:

a. Critical value \(z_{\alpha}=2.33\), test statistic \(z_0 = 1.90\), conclusion: Do not reject \(H_0\). There is insufficient evidence that \(p>0.23\) (Option A)
b. \(p - value=0.029\)