Question

2) You have been asked to study whether there are any differences in the proportions of stores that have experienced a return rate of the shirt over the recent period of at least 20% based upon the region of the country where the stores are located. Random samples of stores in four regions of the country are selected. The sample data concerning whether the store has experienced a return rate of at least 20% for the item over the given time-period is shown in appendix two below. At each of the 5% and 2% levels of significance, are there any differences in the proportions of the stores in the various parts of the country concerning their having experienced at least a 20% return of the item over the period that is being studied? If at either significance level you find any differences in the proportions of the stores based upon location that have experienced at lease this return rate, perform the necessary work to ascertain where any possible differences lie. Appendix Two: Region ≥20% return One Two Three Four Yes 28 37 29 43 No 52 43 66 47

Answer 1

In the given question ,we are required to

1.Check if the proportions of two samples given are significantly different (4 regions)?

The level of significance being 5% and 2 % each .

This is a case of hypothesis testing of checking of difference between two sample proportions.

The sample proportions in two scenario (samples for the four regions 91,2,3,4) are :-

	Regions
	1	2	3	4
Yes	28	37	29	43
No	52	43	66	47

Now we shall perform hypothesis testing on each region.

Kindly note that we are given with percentage proportions in each case only . Hence we shall take into account that the sample size in each case is 100.

For sample 1, we have that the sample size is N-. 100, the number of favorable cases is X1--28 so then the sample proportion is Pi = = = 0.28 For sample 2, we have that the sample size is N2 100, the number of favorable cases is X252 so then the sample proportion is p10.52 The value of the pooled proportion is computed as00.4 Also, the given significance level is a 0.02 1100 1N 100+100 rna The following null and alternative hypotheses need to be tested This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted (2) Rejection Region Based on the information provided, the significance level is 0.02, and the critical value for a right-tailed test is 2.05 The rejection region for this right-tailed test is R = {: (3 The z-statistic is computed as follows > 2.05} Pi P2 0.28 0.52 3.464 VP(1 p)(1/ni +1/n2) 0.4.(1 0.4) (1/100 +1/100) (4) Decision about the null hypothesis Since it is observed that3.464 rejected 2.05, it is then concluded that the null hypothesis is not Using the P-value approach: The p-value is p 0.9997, and since p concluded that the null hypothesis is not rejected 0.99970.02, it is It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pi is greater than p2, at the 0.02 significance level.

For sample 1, we have that the sample size is N1 100, the number of favorable cases is X!--28 = 0.28 so then the sample proportion is p,- For sample 2, we have that the sample size is N2 100, the number of favorable cases is X2 52 so then the sample proportion is p100.52 The value of the pooled proportion is computed as p Also, the given significance level is 0.05 끓 = 100-100-04 The following null and alternative hypotheses need to be tested This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted (2) Rejection Region Based on the information provided, the significance level is α-: 0.05, and the critical value for a right-tailed test is 1.64 The rejection region for this right-tailed test is R = > 1.64) The z-statistic is computed as follows Pi P2 0.28-0.52 = 3.464 VP(1 P(1/ni +1/n2) v0.4 (1 0.4) (1/100 1/100) (4) Decision about the null hypothesis Since it is observed that3.464 rejected 1.64, it is then concluded that the null hypothesis is not Using the P-value approach. The p-value is p = 0.0997, and since p concluded that the null hypothesis is not rejected 0.9997 > 0.05, it is It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pi is greater than p2, at the 0.05 significance level.

For sample 1, we have that the sample size is N 100, the number of favorable cases is X1 37 so then the sample proportion is pi = 1 = 0.37 For sample 2, we have that the sample size is N2 100, the number of favorable cases is X2- 43 so then the sample proportion is p0 0.43 The value of the pooled proportion is computed as0 0.4 Also, the given significance level is a 0.02 - 100 100+1 The following null and alternative hypotheses need to be tested Ho:p1 =p2 This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted (2) Rejection Region Based on the information provided, the significance level is α-0.02, and the critical value for a right-tailed test is 2.05 The rejection region for this right-tailed test is R 2.05 (3 The z-statistic is computed as follows 0.37-0.43 0.866 V(1 -P(1/ni +1/n2) V0.4 (1 0.4) (1/100 1/100) (4) Decision about the null hypothesis Since it is observed that0.866 2.05, it is then concluded that the null hypothesis is not rejected Using the P-value approach: The p-value is p = 0.8068, and since p = 0.8068 > 0.02, it is concluded that the null hypothesis is not rejected It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pi is greater than p2, at the 0.02 significance level

For sample 1, we have that the sample size is N-. 100, the number of favorable cases is X1 37 so then the sample proportion is p100 0.37 For sample 2, we have that the sample size is N2 100, the number of favorable cases is X243 so then the sample proportion is p0 0.43 The value of the pooled proportion is computed as10 0.4 Also, the given significance level is a 0.05 100 37+43 1N 100+100 rna The following null and alternative hypotheses need to be tested This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted (2) Rejection Region Based on the information provided, the significance level is 0.05, and the critical value for a right-tailed test is 1.64 The rejection region for this right-tailed test is R-{: ะ > 1.64) (3 The z-statistic is computed as follows Pi P2 0.37 0.43 0.866 VP(1 p)(1/ni +1/n2) 0.4.(1 0.4) (1/100 +1/100) (4) Decision about the null hypothesis Since it is observed that0.866 rejected 1.64, it is then concluded that the null hypothesis is not Using the P-value approach: The p-value is p 0.8068, and since p concluded that the null hypothesis is not rejected 0.80680.05, it is It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pi is greater than p2, at the 0.05 significance level.

For sample 1, we have that the sample size is N 100, the number of favorable cases is X1- 29 so then the sample proportion is p.- =刕= 0.29 For sample 2, we have that the sample size is N2 100, the number of favorable cases is X2 66, so then the sample proportion is 0.66 The value of the pooled proportion is computed as p =-= 100-100-0475 Also, the given significance level is a 0.02 The following null and alternative hypotheses need to be tested This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted (2) Rejection Region Based on the information provided, the significance level is α-_ 0.02, and the critical value for a right-tailed test is:c-2.05 The rejection region for this right-tailed test is R = {: > 2.05) (3 The z-statistic is computed as follows Pi P2 0.29 0.66 5.239 I-P)/n+1/20 0.475 (1 0.475) (1/100+1/100) Since it is observed that5.239 2.05, it is then concluded that the null hypothesis is not rejected Using the P-value approach: The p-value is p 1, and since p 10.02, it is concluded that the null hypothesis is not rejected (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pi is greater than p2, at the 0.02 significance level

For sample 1, we have that the sample size is Ni :-100, the number of favorable cases is X1 : 29 so then the sample proportion is pT0.29 For sample 2, we have that the sample size is N2 100, the number of favorable cases is X2 66 so then the sample proportion is p0.66 The value of the pooled proportion is computed as p = 는 100-100-0475 Also, the given significance level is 0.05 The following null and alternative hypotheses need to be tested Ho:p1 = P2 This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted (2) Rejection Region Based on the information provided, the significance level is α 0.05, and the critical value for a right-tailed test is 1.64 The rejection region for this right-tailed test is R = {: > 1.64 The z-statistic is computed as follows PiP2 0.29 0.66 5.239 Vp(1 -P)(1/n +1/n2) 0.475. (1 - 0.475)(1/100 +1/100) Since it is observed that5.239 1.64, it is then concluded that the null hypothesis is not rejected. Using the P-value approach. The p-value is p = 1, and since p = 1 > 0.05, it is concluded that the null hypothesis is not rejected (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pi is greater than p2, at the 0.05 significance level.

For sample 1, we have that the sample size is N1 100, the number of favorable cases is XI-: 43 so then the sample proportion is 0.43 For sample 2, we have that the sample size is N2 100, the number of favorable cases is X2- 47, so then the sample proportion is p000.47 The value of the pooled proportion is computed as- 100 0.45 Also, the given significance level is α 0.02 43+47 100+1 The following null and alternative hypotheses need to be tested Ho P2 This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted (2) Rejection Region Based on the information provided, the significance level is α-0.02, and the critical value for a right-tailed test is 2.05 The rejection region for this right-tailed test is R 2.05 The z-statistic is computed as follows Pi P2 0.43-0.47 --0.569 VP(1 P(1/ni +1/n2) V0.45 (1-0.45) (1/100 +1/100) (4) Decision about the null hypothesis Since it is observed that0.569 2.05, it is then concluded that the null hypothesis is not rejected Using the P-value approach. The p-value is p = 0.7152, and since p = 0.7152 > 0.02, it is concluded that the null hypothesis is not rejected It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pi is greater than p2, at the 0.02 significance level.

For sample 1, we have that the sample size is N 100, the number of favorable cases is Xi 43 so then the sample proportion is p.-# = -0.43 For sample 2, we have that the sample size is N2 100, the number of favorable cases is X2 47, so then the sample proportion is 00.47 The value of the pooled proportion is computed as p 100-100 045 Also, the given significance level is a0.05 43447 The following null and alternative hypotheses need to be tested This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted (2) Rejection Region Based on the information provided, the significance level is 0.05, and the critical value for a right-tailed test is 1.64 The rejection region for this right-tailed test is R = {: > 1.64) The z-statistic is computed as follows Pi P2 0.43 0.47 = 0.569 V(1 P(1/ni +1/n2) V0.45 (1 0.45)(1/100+1/100) (4) Decision about the null hypothesis Since it is observed that0.569 rejected 1.64, i is then concluded that the null hypothesis is not Using the P-value approach. The p-value is p 0.7152, and since p = 0.7152 > 0.05, it is concluded that the null hypothesis is not rejected It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pi is greater than p2, at the 0.05 significance level