Question

you wish to test the following claim ($h_a$) at a significance level of $alpha = 0.02$.
$h_0:p_1 = p_2$
$h_a:p_1
eq p_2$
the 1st populations sample has 134 successes and a sample size = 671.
the 2nd populations sample has 57 successes and a sample size = 475.
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...
○ reject the null
○ accept the null
○ fail to reject the null
as such, the final conclusion is that...
○ the sample data support the alternate hypothesis claim that p1 is not equal to p2.
○ there is not sufficient sample evidence to support the alternate hypothesis claim that p1 is not equal to p2.

Explanation:

Step1: Calculate sample proportions

Let $\hat{p}_1=\frac{x_1}{n_1}$, where $x_1 = 134$ and $n_1=671$. So, $\hat{p}_1=\frac{134}{671}\approx0.1997$. Let $\hat{p}_2=\frac{x_2}{n_2}$, where $x_2 = 57$ and $n_2 = 475$. So, $\hat{p}_2=\frac{57}{475}=0.12$.

Step2: Calculate pooled proportion

$\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}=\frac{134+57}{671 + 475}=\frac{191}{1146}\approx0.1667$.

Step3: Calculate the test - statistic

The formula for the $z$ - test statistic for two - proportion 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:

$$
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Step4: Calculate the p - value

Since this is a two - tailed test, the $p$ - value is $2\times(1 - P(Z<|z|))$.
From the standard normal table, $P(Z < 3.558)\approx0.9998$.
So, $p - value=2\times(1 - 0.9998)=0.004$.

Step5: Make a decision

Since $\alpha = 0.02$ and $p - value=0.004<0.02$, we reject the null hypothesis.
Since we reject the null hypothesis, the sample data support the alternate hypothesis claim that $p_1
eq p_2$.

Answer:

test statistic = 3.558
p - value = 0.004
The p - value is less than (or equal to) $\alpha$
This test statistic leads to a decision to reject the null
As such, the final conclusion is that The sample data support the alternate hypothesis claim that $p_1$ is not equal to $p_2$.