1. BCOR 1020 Business Statistics Lecture 22 – April 10, 2008

2. Overview Chapter 10 – Two-Sample Tests –Two-Sample Tests –Comparing Two Proportions –Comparing Two Means (  known)

3. Chapter 10 – Two-Sample Tests A Two-sample test compares two sample estimates with each other. A one-sample test compares a sample estimate against a non-sample benchmark. What is a Two-Sample Test? Basis of Two-Sample Tests: Two samples that are drawn from the same population may yield different estimates of a parameter due to chance.

4. Chapter 10 – Two-Sample Tests If the two sample statistics differ by more than the amount attributable to chance, then we conclude that the samples came from populations with different parameter values. What is a Two-Sample Test?

5. Chapter 10 – Two-Sample Tests Choose Level of Significance, . State the hypotheses. Define the test statistic and its distribution under H 0. Set up the decision rule. Insert the sample statistics. Make a decision based on the critical values or using p-values. If our decision is wrong, we could commit a type I or type II error. Larger samples are needed to reduce type I or type II errors. Test Procedure:

6. Chapter 10 – Comparing Two Proportions To compare two population proportions,  1,  2, use the following hypotheses… Testing for Zero Difference:  1 =  2

7. Chapter 10 – Comparing Two Proportions The sample proportion p 1 is a point estimate of  1 : Testing for Zero Difference:  1 =  2 The sample proportion p 2 is a point estimate of  2 :

8. Chapter 10 – Comparing Two Proportions If H 0 is true, there is no difference between  1 and  2, so the samples are pooled (averaged) into one “big” sample to estimate the common population proportion. Pooled Proportion: = number of successes in combined samples combined sample size p = x 1 + x 2 n 1 + n 2

9. Chapter 10 – Comparing Two Proportions If the samples are large, p 1 – p 2 may be assumed normally distributed. The test statistic is the difference of the sample proportions divided by the standard error of the difference. The standard error is calculated by using the pooled proportion. The test statistic for the hypothesis  1 =  2 is: Test Statistic: ~N(0,1) under H 0.

10. Chapter 10 – Comparing Two Proportions The test statistic for the hypothesis  1 =  2 may also be written as: Test Statistic: ~N(0,1) under H 0.

11. Chapter 10 – Comparing Two Proportions Check the normality assumption with n  > 10 and n(1-  ) > 10. –Each of the two samples must be checked separately using each sample proportion in place of . –If either sample proportion is not normal, their difference cannot safely be assumed normal. –The sample size rule of thumb is equivalent to requiring that each sample contains at least 10 “successes” and at least 10 “failures.” Checking Normality: If sample sizes do not justify the normality assumption, treat each sample as a binomial experiment. If the samples are small, the test is likely to have low power. Small Samples:

12. Chapter 10 – Comparing Two Proportions Steps in Testing Two Proportions: If we wish to test a hypothesis about the equality of two population proportions, we will follow the same logic… –Specify the level of significance,  (given in problem or assume 5% or 10%). –State the null and alternative hypotheses, H 0 and H 1 (based on the problem statement). –Compute the test statistic and determine its distribution under H 0. –State the decision criteria (based on the hypotheses and distribution of the test statistic under H 0 ). –State your decision.

13. Chapter 10 – Comparing Two Proportions Example: A purchasing agent for a music production company is trying to find a CD vendor. In order to distinguish between he two lowest bidders, random samples of CDs from both vendors are compared for quality – using the proportion of CDs that are defective as the measure of quality. In a sample of 200 CDs from vendor A, 18 are found to be defective. In a sample of 300 CDs from vendor B, 18 are found to be defective. Test the appropriate hypothesis to determine whether there is any difference in quality between the two vendors.

14. Chapter 10 – Comparing Two Proportions Example: The sample proportions of defective CDs for both vendors are… p A = 18/200 = 0.09p B = 18/300 = 0.06 The pooled estimate is… = 36/500 = 0.072 Testing the hypothesis to determine whether there is any difference in quality between the two vendors gives us… p = x 1 + x 2 n 1 + n 2

15. Clickers Using the formula compute the test statistic for this hypothesis test. (A) Z* = 1.148 (B) Z* = 1.384 (C) Z* = 1.271 (D) Z* = 1.645

16. Chapter 10 – Comparing Two Proportions Example: We will reject H 0 in favor of H 1, if the absolute value of our test statistic exceeds Z  /2 = Z.05 = 1.645. So, we fail to reject H 0. At the 10% level, there is not statistically significant evidence that the proportion of defects is different for the two vendors.

17. Clickers Using the 2-Sided formula … compute the p-value for this hypothesis test. (A) p-value = 0.1000 (B) p-value = 0.1020 (C) p-value = 0.1437 (D) p-value = 0.2040

18. Chapter 10 – Comparing Two Means: Indep. Samples (  known) The hypotheses for comparing two independent population means  1 and  2 are: Format of Hypotheses: Case 1: If the population variances  1 2 and  2 2 are known, then use the normal distribution. Test Statistic:

19. Chapter 10 – Comparing Two Means: Indep. Samples (  known) Decision Criteria: Make a decision based on the critical values of the Z distribution or using p-values.

20. Chapter 10 – Comparing Two Means: Indep. Samples (  known) Example: Suppose we are selecting between two shipping companies. A random sample of 10 shipments from company 1 had an average of 5.4 days while a random sample of 12 shipments from company 2 had an average of 6.2 days. If the standard deviation for shipping times from both companies is known to be  1 =  2 = 1.5 days, test the appropriate hypothesis to determine whether there is a difference in average shipping times between these companies. (Use a 10% level of significance.) (overhead)

21. Chapter 10 – Comparing Two Means: Indep. Samples (  known) Example (continued): We know…  1 =  2 = 1.5; n 1 = 10; n 2 = 12; Based on the problem statement we select the two- sided test …

22. Clickers How is our test statistic Z* = -1.25 distributed if H 0 is true? (A) Student’s t with 21 d.f. (B) Standard Normal (C) Normal with  = 5.4,  = 1.5 (D) Not enough information

23. Chapter 10 – Comparing Two Means: Indep. Samples (  known) Example (continued): Rejection Criteria… Reject H 0 in favor of H 1 if |Z*| > Z a/2 = Z.10/2 = Z.05 = 1.645 Decision… Since |Z*| is less than Z a/2, we fail to reject H 0. There is not statistically significant evidence of a difference in mean shipping times.