Question

A sample of 400 items produced by supplier A contained 70 defective items. A sample of 300 items produced by supplier B contained 40 defective items.

a) (5 points) Using the critical value approach, test at 4% level of significance, if there is a difference between the true proportion of defective items for the two suppliers.

b) (3 points) Compute the p-value associated with test in (a) and test the hypothesis in (a).

c) (3 points) Construct a 96% confidence interval for the difference between true defective proportions and test the hypothesis in (a).

Answer 1

For sample 1, we have that the sample size is N1 = 400, the number of favorable cases is Xı = 70, so then the sample proportion is fi = x 70 = 0.175 = 400 For sample 2, we have that the sample size is N2 = 300, the number of favorable cases is X, = 40, so then the sample proportion is P2 = x; 40 = 0.1333 300 The value of the pooled proportion is computed as p = x + x = 70+40 400+300 = 0.1571 Also, the given significance level is a = 0.04. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p1 = P2 Ha: Py = P2 This corresponds to a two-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 a = 0.04, and the critical value for a two-tailed test is ze=2.05. The rejection region for this two-tailed test is R = {z: 21 > 2.05} (3) Test Statistics The Z-statistic is computed as follows: pi - P2 V (1-7)(1/ni + 1/n2) 0.175 – 0.1333 0.1571 .(1 – 0.1571)(1/400 +1/300) = 1.499 (4) Decision about the null hypothesis Since it is observed that 2 = 1.499 <% = 2.05, it is then concluded that the null hypothesis is not rejected. Using the P-value approach: The p-value is p = 0.1339, and since p= 0.1339 > 0.04, 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 pı is different than P2, at the 0.04 significance level. Confidence Interval The 96% confidence interval for Pi - p2 is: -0.014<P - P2<0.098.

As confidence interval contains 0, we conclude that the there is no significant differences between the proportion.

Dear student,
I am waiting for your feedback. I have given my 100% to solve your queries. If you satisfied with my answer then please please like this.
Thank You