You wish to test the following claim $(H_{a})$ at a significance level of $α = 0.05$
$\begin{array}{l} H_{o} : p_{1} = p_{2} \\ H_{a} : p_{1} \neq p_{2} \end{array}$
You obtain $45.2 %$ successes in a sample of size $n_{1} = 531$ from the first population. You obta successes in a sample of size $n_{2} = 657$ from the second population.

What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic $=$
What is the p-value for this sample? (Report answer accurate to four decimal places.) $p - value =$ The $p$ -value is...
less than (or equal to) $α$
greater than $α$
$0^{6}$
This test statistic leads to a decision to...
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\text{This test statistic leads to a decision to reject the null hypothesis.}

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Step 1 ： $X_{1} = 0.452 n_{1} = 240$

Step 2 ： $X_{2} = n_{2} - n_{1} X_{1} = 657 - 240 = 417$

Step 3 ： $\hat{p} = \frac{X_{1} + X_{2}}{n_{1} + n_{2}} = \frac{240 + 417}{531 + 657} = 0.5040$

Step 4 ： ${\hat{p}}_{1} = \frac{X_{1}}{n_{1}} = \frac{240}{531} = 0.4520$

Step 5 ： ${\hat{p}}_{2} = \frac{X_{2}}{n_{2}} = \frac{417}{657} = 0.6347$

Step 6 ： $test statistic = \frac{{\hat{p}}_{1} - {\hat{p}}_{2}}{\sqrt{\frac{\hat{p} (1 - \hat{p})}{n_{1}} + \frac{\hat{p} (1 - \hat{p})}{n_{2}}}} = \frac{0.4520 - 0.6347}{\sqrt{\frac{0.5040 (1 - 0.5040)}{531} + \frac{0.5040 (1 - 0.5040)}{657}}} = - 6.363$

Step 7 ： $p-value = 2 * P (Z \leq - 6.363) = 2 * (1 - P (Z \leq 6.363)) \approx 2 * 0 = 0.0000$

Step 8 ：\text{The p-value is less than } $α$

Step 9 ：\text{This test statistic leads to a decision to reject the null hypothesis.}

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