Question

Conduct a test at the alphaequals0.05 level of significance by determining ​(a) the null and alternative​ hypotheses, ​(b) the test​ statistic, and​ (c) the​ P-value. Assume the samples were obtained independently from a large population using simple random sampling. Test whether p 1 greater than p 2. The sample data are x 1 equals 125​, n 1 equals 247​, x 2 equals 132​, and n 2 equals 312. ​(a) Choose the correct null and alternative hypotheses below. A. Upper H 0 : p 1 equals p 2 versus Upper H 1 : p 1 less than p 2 B. Upper H 0 : p 1 equals p 2 versus Upper H 1 : p 1 greater than p 2 C. Upper H 0 : p 1 equals 0 versus Upper H 1 : p 1 not equals 0 D. Upper H 0 : p 1 equals p 2 versus Upper H 1 : p 1 not equals p 2 ​(b) Determine the test statistic. z0equals nothing ​(Round to two decimal places as​ needed.) ​(c) Determine the​ P-value. The​ P-value is nothing. ​(Round to three decimal places as​ needed.) What is the result of this hypothesis​ test? A. Do not reject the alternative hypothesis because there is sufficient evidence to conclude that p 1 not equals p 2. B. Reject the null hypothesis because there is sufficient evidence to conclude that p 1 greater than p 2. C. Reject the null hypothesis because there is sufficient evidence to conclude that p 1 less than p 2. D. Do not reject the null hypothesis because there is not sufficient evidence to conclude that p 1 less than p 2. Click to select your answer(s).

Answer 1

For sample 1, we have that the sample size is N1​=247, the number of favorable cases is X1​=125, so then the sample proportion is

For sample 2, we have that the sample size is N2​=312, the number of favorable cases is X2​=132, so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.05.

(a) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:

Ha: ​

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

Rejection Region

Based on the information provided, the significance level is α=0.05, and the critical value for a right-tailed test is zc​=1.64.

The rejection region for this right-tailed test is R={z:z>1.64}

(b)Test Statistics

The z-statistic is computed as follows:

(c)

The p-value is p = 0.0253

(d) Decision about the null hypothesis

B. Reject the null hypothesis because there is sufficient evidence to conclude that p1 greater than p2

Conceptual Knowledge

Since it is observed that z=1.955>zc​=1.64, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: p=0.0253, and since p=0.0253<0.05, it is concluded that the null hypothesis is rejected.

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is greater than p2​, at the 0.05 significance level.

Graphically

Please let me know in comments if anything is not clear. Will reply ASAP. Please do upvote if satisfied!!