Question

Independent random samples, each containing 900 observations, were selected from two binomial populations. The samples from populations 1 and 2 produced 254 and 658 successes, respectively.

(a) Test H0:(p1−p2)=0 against Ha:(p1−p2)≠0. Use α=0.07

test statistic =

rejection region |z|>

The final conclustion is:_______

A. There is not sufficient evidence to reject the null hypothesis that (p1−p2)=0.
B. We can reject the null hypothesis that (p1−p2)=0 and accept that (p1−p2)≠0.

(b) Test H0:(p1−p2)=0 against Ha:(p1−p2)>0. Use α=0.01
test statistic =
rejection region z>

Answer 1

To test the null hypothesis H0: (p1-p2) = 0 against the alternative hypothesis Ha: (p1-p2) ≠ 0, we can use the two-sample z-test for comparing population proportions. The test statistic formula is z = (p1 - p2) / sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2)). With sample proportions of p1 = 0.2822 and p2 = 0.7311, and a test statistic |z| = 8.9731, which is greater than the critical value ±1.81, we reject the null hypothesis and accept the alternative hypothesis.

Step-by-step explanation:

To test the null hypothesis H0: (p1-p2) = 0 against the alternative hypothesis Ha: (p1-p2) ≠ 0, we can use the two-sample z-test for comparing population proportions. The test statistic formula is:

z = (p1 - p2) / sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2))

Where p1 and p2 are the sample proportions, n1 and n2 are the sample sizes, and sqrt refers to the square root.

In this case, the sample proportions are p1 = 254/900 = 0.2822 and p2 = 658/900 = 0.7311. With sample sizes of 900 for both populations and an α value of 0.07, the critical z-value is ±1.81 (obtained using a Z-table or calculator).

Comparing the test statistic |z| = |-8.9731| = 8.9731 with the critical value ±1.81, we find that |z| > 1.81. Therefore, we reject the null hypothesis H0 and conclude that there is sufficient evidence to support the alternative hypothesis Ha: (p1-p2) ≠ 0. The correct answer is B. We can reject the null hypothesis that (p1-p2) = 0 and accept that (p1-p2) ≠ 0.