Question

test the hypothesis using the p - value approach. be sure to verify the requirements of the test
$h_0: p = 0.8$ versus $h_1: p>0.8$
$n = 240$, $x = 187$, $\alpha=0.05$
because $np_0(1 - p_0)=\square\square10$, the sample size is 5% of the population size, and the parents in the sample selected at random, all of the requirements for testing the hypothesis about the population proportion satisfied
(type an integer or a decimal. do not round)
compute the test statistic, $z_0$
$z_0=\square$
(round to two decimal places as needed)
compute the p - value
p - value $=\square$
(round to three decimal places as needed)
draw a conclusion. choose the correct answer.
a. reject the null hypothesis, because the p - value is greater than $\alpha$. there is sufficient evidence to conclude that $p>0.8$
b. reject the null hypothesis, because the p - value is less than $\alpha$. there is sufficient evidence to conclude that $p>0.8$
c. do not reject the null hypothesis, because the p - value is less than $\alpha$. there is insufficient evidence to conclude that $p>0.8$
d. do not reject the null hypothesis, because the p - value is greater than $\alpha$. there is insufficient evidence to conclude that $p>0.8$

Explanation:

Step1: Calculate \(np_0(1 - p_0)\)

Given \(n = 240\), \(p_0=0.8\)

$$np_0(1 - p_0)=240\times0.8\times(1 - 0.8)=240\times0.8\times0.2 = 38.4$$

Since \(38.4>10\), the normal approximation to the binomial is valid.

Step2: Calculate the sample proportion \(\hat{p}\)

\(\hat{p}=\frac{x}{n}\), with \(x = 187\) and \(n = 240\)
\(\hat{p}=\frac{187}{240}\approx0.779\)

Step3: Calculate the test statistic \(z_0\)

The formula for the test statistic in a one - sample proportion test is \(z_0=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\)
Substitute \(\hat{p}=0.779\), \(p_0 = 0.8\), \(n = 240\)

$$z_0=\frac{0.779 - 0.8}{\sqrt{\frac{0.8\times(1 - 0.8)}{240}}}=\frac{-0.021}{\sqrt{\frac{0.16}{240}}}=\frac{-0.021}{\sqrt{0.000667}}\approx\frac{-0.021}{0.0258}\approx - 0.80$$

Step4: Calculate the \(P\) - value

Since \(H_1:p>0.8\), the \(P\) - value is \(P(Z > z_0)\)
We know that \(P(Z>z_0)=1 - P(Z\leq z_0)\)
From the standard normal table, for \(z_0=-0.80\), \(P(Z\leq - 0.80)=0.2119\)
So \(P - value=1 - 0.2119=0.788\)

Step5: Make a decision

Given \(\alpha = 0.05\)
Since \(P - value=0.788>0.05=\alpha\), we reject the null hypothesis. There is sufficient evidence to conclude that \(p > 0.8\)

Answer:

• \(np_0(1 - p_0)=240\times0.8\times(1 - 0.8)=38.4\)
• \(z_0=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\), where \(\hat{p}=\frac{x}{n}=\frac{187}{240}\approx0.779\), \(z_0=\frac{0.779 - 0.8}{\sqrt{\frac{0.8\times(1 - 0.8)}{240}}}\approx - 0.80\)
• \(P - value = P(Z>z_0)\) (since \(H_1:p > 0.8\)), \(P(Z>- 0.80)=1 - P(Z\leq - 0.80)\)

From the standard - normal table, \(P(Z\leq - 0.80)=0.2119\), so \(P - value=1 - 0.2119 = 0.788\)

• \(A\). Reject the null hypothesis, because the \(P\) - value is greater than \(\alpha\). There is sufficient evidence to conclude that \(p>0.8\)