Question

test the hypothesis using the p - value approach. be sure to verify the requirements of the test.
$h_0: p = 0.8$ versus $h_1: p > 0.8$
$n = 250, x = 205, \alpha = 0.05$

is $np_0(1 - p_0) \geq 10$?
\bigcirc yes
\bigcirc no

use technology to find the p - value.
p - value = \square
(round to three decimal places as needed.)
\square the null hypothesis, because the p - value is \square than $\alpha$

Explanation:

Step1: Check test requirements

Given \( n = 250 \), \( p_0 = 0.8 \). Calculate \( np_0(1 - p_0) \):
\( np_0(1 - p_0)=250\times0.8\times(1 - 0.8)=250\times0.8\times0.2 = 40 \).
Since \( 40\geq10 \), the answer to "Is \( np_0(1 - p_0)\geq10 \)?" is Yes.

Step2: Calculate sample proportion

Sample proportion \( \hat{p}=\frac{x}{n}=\frac{205}{250}=0.82 \).

Step3: Compute test statistic

For a proportion test, the test statistic \( z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \).
Substitute values:
\( z=\frac{0.82 - 0.8}{\sqrt{\frac{0.8\times0.2}{250}}}=\frac{0.02}{\sqrt{\frac{0.16}{250}}}=\frac{0.02}{\sqrt{0.00064}}\approx\frac{0.02}{0.0253}\approx0.791 \).

Step4: Find P - value

For \( H_1:p > 0.8 \) (right - tailed test), the P - value is \( P(Z>z) \) where \( Z \) is standard normal. Using technology (e.g., calculator or software), \( P(Z > 0.791)\approx0.214 \) (rounded to three decimal places).

Step5: Make decision

Compare P - value (\( 0.214 \)) with \( \alpha = 0.05 \). Since \( 0.214>0.05 \), we fail to reject the null hypothesis.

Answer:

• Is \( np_0(1 - p_0)\geq10 \)? Yes
• P - value \( = 0.214 \)
• We fail to reject the null hypothesis, because the P - value is greater than \( \alpha \).