Hypothesis Test for the Difference in Two Proportions
You wish to test the following claim $(H_{d})$ at a significance level of $α = 0.02$ .
$\begin{array}{l} H_{o} : p_{1} = p_{2} \\ H_{a} : p_{1} \neq p_{2} \end{array}$
You obtain 208 successes in a sample of size $n_{1} = 291$ from the first population. in a sample of size $n_{2} = 263$ from the second population.

What is the test statistic for this sample? (Report answer accurate to three decima 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 $α$

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5. Compare the $p$ -value with the significance level $α = 0.02$

Steps

Step 1 ：1. Calculate $\hat{p_{c}}$ and $\hat{p_{1}}$ and $\hat{p_{2}}$ : $\hat{p_{c}} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}} = \frac{208}{291 + 263} \hat{p_{1}} = \frac{208}{291} \hat{p_{2}} = \frac{208 - 208}{263}$

Step 2 ：2. Calculate the standard error: $S E = \sqrt{\frac{\hat{p_{c}} (1 - \hat{p_{c}})}{n_{1}} + \frac{\hat{p_{c}} (1 - \hat{p_{c}})}{n_{2}}}$

Step 3 ：3. Calculate the test statistic: $Z = \frac{(\hat{p_{1}} - \hat{p_{2}}) - 0}{S E}$

Step 4 ：4. Find the $p$ -value by using the standard normal table with Z

Step 5 ：5. Compare the $p$ -value with the significance level $α = 0.02$

Answer

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