Conduct the following test at the ? = 0.05 level of...

Conduct the following test at the ? = 0.05 level of significance by determining (a) the null and alternative hypotheses, (b) the test statistic, and (c) the critical value. Assume that the samples were obtained independently using simple random sampling. Test whether p1 ? p2. Sample data are x1 = 28, n1 = 254, x2 = 36 and n2 = 301. (a) Determine the null and alternative hypotheses. Choose the correct answer below. H0: p1 = p2 versus H1: p1 ? p2 H0: p1 = p2 versus H1: p1 < p2 H0: p1 = p2 versus H1: p1 > p2 (b) The test statistic z0 is . (Round to two decimal places as needed.) (c) The critical values are ± . (Round to three decimal places as needed.) Test the null hypothesis. Choose the correct conclusion below. Do not reject the null hypothesis. Reject the null hypothesis.

Transcript

00:01 In the given question we are told that we should conduct the following test at a level of significance of 0 .05, 0 .05.

00:15 We should determine the null and alternative hypothesis, the test statistic, the critical value.

00:25 So these are the things that we are told to find.

00:27 And the data given to us is to test whether the proportions p1 and p2 are equal.

00:39 So the data is the first sample is having x1 to be equal to 22 and the sample size n1 is 254.

00:52 So to find the proportion from this data we can take 28 divided by 252.

00:58 And this would give us the proportion p1 to be equal to 0 .11 and the second group is x2 which is equal to 36 n1 which is the n2 which is the second the sample size of this the second sample which is given as 301 and so we can take p2 over here and as 36 divided by 301 which would be equal to 0 .12 so this is 0 .12 so to take the proportions in four decimal places b1 would be 112 and p2 would be 1901 would be 1901 1916 19 16 19 16 right 0.

02:00 1196 so these are the values of p1 and p2 so the next thing that we are going to do is the to state the null and alternative hypothesis right so the null hypothesis is always a claim that the two entities that is given in the question are equal or they doesn't have any difference in between them so it would be p1 as equal to p2 so the alternate hypothesis would be the one that says p1 is not equal to p2 and now the next thing that we are told to find is the test statistic which we can take as the z value or the z score so there is a formula to find the z value over here which is p1 minus p2 divided by the square root of p cap times one minus p2 p cap times 1 by n1 plus 1 by n2 so we already know p1 and p2 are and n1 and n2 are p cap is what we call as the pooled proportion so the pooled proportion consists of both p1 and p2 and it is found using a formula which is x1 plus x2 divided by n1 plus n2 n1 plus n2 so using this formula we can write p cap is then equal to 28 plus 36 right 28 plus 36 divided by 254 plus 30 and 1001 so this would use a proportion of 0 .11, 0 .1153.

04:14 We have the value of p cap.

04:16 We can also find the value of 1 minus p cap, which would be equal to 0 .847.

04:27 So on substituting these values in this formula, we will have 0 .1102 minus.

04:35 0 .1196 divided by the square root of 0 .153 times 0 .8847 times 1 by n1 plus 1 by n2 would be 1 by 254 plus 1 by 301...

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