Question

test the hypothesis using the p - value approach. be sure to verify the requirements of the test

( h _ { 0 } : p = 0.91 ) versus ( h _ { 1 } : p
eq 0.91 )
( n = 500, x = 450, alpha = 0.01 )

is ( n p _ { 0 } ( 1 - p _ { 0 } ) geq 10 )? select the correct choice below and fill in the answer box to complete your choice.
(type an integer or a decimal. do not round.)

o a. yes, because ( n p _ { 0 } ( 1 - p _ { 0 } ) = )

o b. no, because ( n p _ { 0 } ( 1 - p _ { 0 } ) = )

now find ( hat { p } )

( hat { p } = ) (type an integer or a decimal. do not round.).

find the test statistic ( z _ { 0 } )

( z _ { 0 } = ) (round to two decimal places as needed.)

find the p - value.

p - value ( = ) (round to three decimal places as needed.)

state the conclusion of the hypothesis test

because the p - value is than ( alpha )

Explanation:

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

Given \(n = 500\), \(p_0=0.91\)

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Since \(40.95\geq10\), we choose option A.

Step2: Calculate \(\hat{p}\)

The formula for \(\hat{p}=\frac{x}{n}\), where \(x = 450\), \(n = 500\)

$$ \hat{p}=\frac{450}{500}=0.9 $$

Step3: Calculate the test - statistic \(z_0\)

The formula for \(z_0=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}\)
Substitute \(\hat{p}=0.9\), \(p_0 = 0.91\), \(n = 500\)

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Step4: Calculate the \(P -\)value

Since \(H_1:p
eq0.91\), this is a two - tailed test.
\(P - value=2P(Z\lt z_0)\) (where \(Z\) is a standard normal random variable)
\(P(Z\lt - 1.57)=0.0582\) (from standard normal table)
\(P - value=2\times0.0582 = 0.1164\approx0.117\)

Step5: State the conclusion

Since \(P - value=0.117\) and \(\alpha = 0.01\), and \(0.117>0.01\)
We do not reject \(H_0\) because the \(P -\)value is greater than \(\alpha\)

Answer:

A. Yes, because \(np_0(1 - p_0)=40.95\)
\(\hat{p}=0.9\)
\(z_0=-1.57\)
\(P - value = 0.117\)
Do not reject \(H_0\) because the \(P -\)value is greater than \(\alpha\)