Question

you wish to test the following claim ($h_a$) at a significance level of $alpha = 0.01$. $h_o:p_1 = p_2$ $h_a:p_1>p_2$ the 1st populations sample has 75 successes and a sample size = 375. the 2nd populations sample has 75 successes and a sample size = 632. what is the test statistic (z - score) for this sample? (round to 3 decimal places.) test statistic = what is the p - value for this sample? (round to 3 decimal places.) p - value = the p - value is... greater than $alpha$ less than (or equal to) $alpha$ this test statistic leads to a decision to... fail to reject the null accept the null reject the null as such, the final conclusion is that... there is not sufficient sample evidence to support the alternate hypothesis claim that p1 > p2. the sample data support the alternate hypothesis claim that p1 > p2. question help: message instructor post to forum

Explanation:

Step1: Calculate sample proportions

Let $n_1 = 375$, $x_1=75$, $n_2 = 632$, $x_2 = 75$.
The sample - proportion for the first sample is $\hat{p}_1=\frac{x_1}{n_1}=\frac{75}{375}=0.2$.
The sample - proportion for the second sample is $\hat{p}_2=\frac{x_2}{n_2}=\frac{75}{632}\approx0.119$.
The pooled proportion $\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}=\frac{75+75}{375 + 632}=\frac{150}{1007}\approx0.149$.

Step2: Calculate the z - score

The formula for the z - score in a two - proportion z - test is $z=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$.
Substitute the values:

$$
LATEXBLOCK0
$$

Step3: Calculate the p - value

Since $H_a:p_1>p_2$, the p - value is $P(Z>z)$.
For $z = 3.476$, $P(Z>3.476)=1 - P(Z\leq3.476)$.
From the standard normal table, $P(Z\leq3.476)\approx0.9997$, so the p - value is $1 - 0.9997 = 0.0003\approx0.000$.

Step4: Make a decision

Since the significance level $\alpha = 0.01$ and the p - value ($0.000$) is less than $\alpha$, we reject the null hypothesis.

Answer:

test statistic = 3.476
p - value = 0.000
The p - value is less than (or equal to) $\alpha$
This test statistic leads to a decision to reject the null
The final conclusion is that the sample data support the alternate hypothesis claim that $p1>p2$.