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 = 197, alpha = 0.05 )

because ( n p _ { 0 } left( 1 - p _ { 0 }
ight) = square ) ( 10 ), the sample size is ( 5 % ) of the population size, and the patients 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: Check the requirement \(np_0(1 - p_0)\)

Calculate \(np_0(1 - p_0)=240\times0.8\times(1 - 0.8)=38.4\gt10\)

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

\(\hat{p}=\frac{x}{n}=\frac{197}{240}\approx0.82\)

Step3: Calculate the test statistic \(z_0\)

Use the formula \(z_0=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\)
Substitute \(\hat{p}=0.82\), \(p_0 = 0.8\), \(n = 240\)
\(z_0=\frac{0.82-0.8}{\sqrt{\frac{0.8\times(1 - 0.8)}{240}}}\approx0.77\)

Step4: Calculate the \(P\) - value

Since \(H_1:p\gt0.8\), \(P - value=P(Z\gt z_0)\)
From the standard normal table, \(P(Z\leq0.77)=0.7794\)
\(P - value=1 - 0.7794 = 0.221\)

Step5: Draw a conclusion

Compare \(P\) - value and \(\alpha\). Since \(P - value(0.221)\gt\alpha(0.05)\), do not reject \(H_0\)

Answer:

• \(np_0(1 - p_0)=240\times0.8\times(1 - 0.8)=38.4\gt10\)
• \(z_0=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\), where \(\hat{p}=\frac{x}{n}=\frac{197}{240}\approx0.82\), \(p_0 = 0.8\), \(n = 240\)

\(z_0=\frac{0.82-0.8}{\sqrt{\frac{0.8\times(1 - 0.8)}{240}}}\approx0.77\)

• \(P - value=P(Z\gt z_0)\), \(P(Z\gt0.77)=1 - P(Z\leq0.77)\)

From the standard normal table, \(P(Z\leq0.77)=0.7794\), so \(P - value=1 - 0.7794 = 0.221\)

• D. Do not reject the null hypothesis, because the \(P\) - value is greater than \(\alpha\). There is insufficient evidence to conclude that \(p\gt0.8\)