Question

test the hypothesis using the p - value approach. be sure to verify the requirements of the test
$h_0: p = 0.5$ versus $h_1: p>0.5$
$n = 250$, $x = 135$, $\alpha=0.01$
click here to view page 1 of the table. click here to view page 2 of the table
calculate the test statistic, $z_0$
$z_0=\square$
(round to two decimal places as needed.)
identify the p - value.
$p - value=\square$
(round to three decimal places as needed.)
choose the correct result of the hypothesis test for the p - value approach below
○ a. do not reject the null hypothesis, because the p - value is less than $\alpha$
○ b. reject the null hypothesis, because the p - value is less than $\alpha$
○ c. reject the null hypothesis, because the p - value is greater than $\alpha$
○ d. do not reject the null hypothesis, because the p - value is greater than $\alpha$.

Explanation:

Step1: Calculate the sample proportion \(\hat{p}\)

The sample proportion \(\hat{p}=\frac{x}{n}\), where \(x = 135\) and \(n=250\). So \(\hat{p}=\frac{135}{250}=0.54\)

Step2: Calculate the test statistic \(z_{0}\)

The formula for the test statistic \(z_{0}\) in a one - sample proportion test is \(z_{0}=\frac{\hat{p}-p_{0}}{\sqrt{\frac{p_{0}(1 - p_{0})}{n}}}\), where \(p_{0}=0.5\), \(n = 250\), and \(\hat{p}=0.54\)

Substitute the values:

$$
LATEXBLOCK0
$$

Step3: Calculate the \(P\) - value

Since \(H_{1}:p>0.5\), this is a right - tailed test. The \(P\) - value is \(P(Z>z_{0})\)

For \(z_{0}=1.27\), \(P(Z > 1.27)=1 - P(Z\leqslant1.27)\)

From the standard normal table, \(P(Z\leqslant1.27)=0.8980\)

So \(P - value=1 - 0.8980 = 0.102\)

Step4: Make a decision

We are given \(\alpha=0.01\)

Since \(P - value=0.102>0.01=\alpha\), we do not reject the null hypothesis.

Answer:

1. \(z_{0}=1.27\)
2. \(P - value = 0.102\)
3. D. Do not reject the null hypothesis, because the \(P\) - value is greater than \(\alpha\)