Question

you wish to test the following claim ($h_a$) at a significance level of $alpha = 0.10$. $h_0:p_1 = p_2$ $h_a:p_1 < p_2$ the 1st populations sample has 89 successes and a sample size = 447. the 2nd populations sample has 59 successes and a sample size = 264. 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... accept the null reject the null fail to 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.

Explanation:

Step1: Calculate sample proportions

Let $\hat{p}_1=\frac{x_1}{n_1}$, where $x_1 = 89$ and $n_1=447$, so $\hat{p}_1=\frac{89}{447}\approx0.199$. Let $\hat{p}_2=\frac{x_2}{n_2}$, where $x_2 = 59$ and $n_2 = 264$, so $\hat{p}_2=\frac{59}{264}\approx0.223$. The pooled - proportion $\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}=\frac{89+59}{447+264}=\frac{148}{711}\approx0.208$.

Step2: 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: $\hat{p}_1\approx0.199$, $\hat{p}_2\approx0.223$, $\hat{p}\approx0.208$, $n_1 = 447$, $n_2=264$.

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Step3: Calculate the p - value

Since $H_a:p_1

Step4: Make a decision

Since $\alpha = 0.10$ and p - value $\approx0.223>0.10$, we fail to reject the null hypothesis.

Step5: State the conclusion

There is not sufficient sample evidence to support the alternate hypothesis claim that $p_1

Answer:

test statistic = - 0.762
p - value = 0.223
The p - value is greater than $\alpha$
This test statistic leads to a decision to fail to reject the null
As such, the final conclusion is that There is not sufficient sample evidence to support the alternate hypothesis claim that $p_1 < p_2$.