Question 2
You wish to test the following claim $\left(H_{a}\right)$ at a significance level of $\alpha=0.02$.
\[
\begin{array}{l}
H_{o}: p_{1}=p_{2} \\
H_{a}: p_{1}> p_{2}
\end{array}
\]

You obtain 354 successes in a sample of size $n_{1}=421$ from the first population. You obtain 551 successes in a sample of size $n_{2}=723$ from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic: $z=$
What is the $p$-value for this sample? (Report answer accurate to four decimal places.)
P-value: $p=$

The $\mathrm{p}$-value is...
less than (or equal to) $\alpha$
greater than $\alpha$

This test statistic leads to a decision to...
reject the null
accept the nuil
fail to reject the null

As such, the final conclusion is that...
There is sufficient evidence to warrant rejection of the claim that the first population proportion is greater than the second population proportion.
There is not sufficient evidence to warrant rejection of the claim that the first population proportion is greater than the second population proportion.
The sample data support the claim that the first population proportion is greater than the second population proportion.
There is not sufficient sample evidence to support the claim that the first population proportion is greater than the second population proportion.

Step 1 ：Given values are \(n_1 = 421\), \(x_1 = 354\), \(n_2 = 723\), \(x_2 = 551\), and \(\alpha = 0.02\).

Step 2 ：Calculate the sample proportions \(p_1 = \frac{x_1}{n_1} = \frac{354}{421} = 0.8408551068883611\) and \(p_2 = \frac{x_2}{n_2} = \frac{551}{723} = 0.7621023513139695\).

Step 3 ：Calculate the pooled proportion \(p = \frac{x_1 + x_2}{n_1 + n_2} = \frac{354 + 551}{421 + 723} = 0.791083916083916\).

Step 4 ：Calculate the test statistic \(z = \frac{p_1 - p_2}{\sqrt{p(1 - p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} = \frac{0.8408551068883611 - 0.7621023513139695}{\sqrt{0.791083916083916(1 - 0.791083916083916)\left(\frac{1}{421} + \frac{1}{723}\right)}} = 3.1598446203260133\).

Step 5 ：Calculate the p-value using the survival function (1 - cumulative distribution function) of the normal distribution. The p-value is approximately 0.0007892664843697467.

Step 6 ：Since the p-value is less than the significance level of 0.02, we reject the null hypothesis.

Step 7 ：This means that there is sufficient evidence to warrant rejection of the claim that the first population proportion is equal to the second population proportion.

Step 8 ：In other words, there is sufficient evidence to support the claim that the first population proportion is greater than the second population proportion.

Step 9 ：The final answer is: The test statistic is approximately \(\boxed{3.160}\) and the p-value is approximately \(\boxed{0.0008}\). We reject the null hypothesis, so there is sufficient evidence to support the claim that the first population proportion is greater than the second population proportion.