1. PowerPoint presentations prepared by Lloyd Jaisingh, Morehead State University Statistical Inference: Hypotheses testing for single and two populations Chapters 9 and 10 MSIS 111 Prof. Nick Dedeke

2. Test hypotheses for one population mean using the Z statistic. Test hypotheses and construct confidence intervals about the difference in two population means using the Z statistic. Test hypotheses and construct confidence intervals about the difference in two related populations. Learning Objectives

3. A research hypothesis is a statement about what the researcher believes will be the outcome of an experiment or study. statistical hypothesis To prove a research hypothesis a statistical hypothesis is formulated and tested. Definition of Hypothesis

4. A statistical hypothesis uses the following logic. It assumes that a condition holds (Null hypothesis, Ho) and attempts to use statistical procedures to show that the condition is supported by data as being more likely to be valid than not. If the null hypothesis is rejected, the alternate hypothesis (Ha) is accepted without testing. Definition of Statistical Hypothesis Testing

5. The research hypothesis: Has the weight of our product changed? Statistical hypothesis to test: Example Statistical Hypothesis Testing (Two-tailed Tests)

6. The research hypothesis: Are the groceries prices in department stores higher than in pharmacies? Statistical hypothesis to test: Example Statistical Hypothesis Testing (One-tailed Tests)

7. Reject and Nonreject Regions X Accept region Reject region  40 ounces Critical values Reject region Experiment: We take a sample of 100 units from the population and calculate the average of the weight of the products. We get a value of 40.3. Though this value is not = 40, we will accept the H0 that the population has a mean = 40, if the sample average value falls in the accept region.

8. Type 1 and Type 2 Errors Reject null Null true Null false Probability of committing Type 1 error is . Probability of committing Type 1 error is . Fail to reject Correct decision Type I error (  Type II error (  Null hypothesis is true but we reject it. Null hypothesis is false but we accept it.

9. Example: Determining the Reject and Nonreject Regions Accept region Reject region  40 ounces Critical values Reject region If we set alpha (  ) to be 0.05. The critical values will be at  /2 = 0.025. This means that the from the normal table we find the z value that corresponds to (0.5-0.025) =0.475 This is z = 1.96

10. Exercise: Determining the Reject and Nonreject Regions Accept region Reject region  40 ounces Critical values Reject region If we set alpha (  ) to be 0.10. Find the z values for the critical value.

11. Response: Determining the Reject and Nonreject Regions Accept region Reject region  40 ounces Critical values Reject region If we set alpha (  ) to be 0.10. The critical values will be at  /2 = 0.05. This means that the from the normal table we find the z value that corresponds to (0.5-0.05) =0.45 This is z = 1.64

12. Example Is the weight of the units of a product different from 40 oz.? Standard deviation is 4. We want probability of Type 1 error alpha  = 0.05; n = 50;  oz; x = 40.6 oz. Steps: 1. Determine the critical values. Z critical = =/- 1.96 2. Calculate the z value. 3. Identify the location of the z value in normal graph. Z is less than critical z, so we accept null hypothesis!

13. Response: Graph Accept region Z = 1.06  40 ounces Critical values Reject region If we set alpha (  ) to be 0.05. The critical values will be at  /2 = 0.025. This means that the from the normal table we find the z value that corresponds to (0.5-0.025) =0.475 This is z critical = 1.96 z= 1.06 will be in the accept zone.

14. Exercise Is the salary of CPA’s different from $74,914? Standard deviation is $14,530. We want probability of Type 1 error to be alpha =0.05; n = 212; mean derived from sample is $78,695. Should we accept the null hypothesis?

15. Exercise(One sided hypothesis) The historical mean rating for mangers is 4.30. A survey of 32 mangers yielded a mean of 4.156. Is this mean less than the historical value? Standard deviation is 0.574. We want probability of Type 1 error to be alpha =0.05; Should we accept the null hypothesis? Notice in this case, we have one-sided case, so the reject zone is on the left side. So, we will use alpha = 0.05 to discover the critical value = 0.5 – 0.05 =0.45 From the table z critical = -1.645 Entering values into equation yields z = -1.42 (try this on your own) z critical is less than –1.42 so we accept null hypothesis.

16. Two Populations: Inferences Up till now we considered cases in which one took a single sample and we use it to test a hypothesis. Often, one needs to compare two different samples. The hypothesis of interest are: Are the samples different? Is one sample less than or greater than the other?

17. Two Populations: Inferences Experiment: Select two independent samples calculate the sample means for each of them. Use the differences between the sample means to test the hypothesis that both of the populations are different. The process is the same, however the equations for deriving z values is now different. We also need samples that exceed 30 items to benefit from the central limit theorem.

18. Sampling Distribution of the Difference Between Two Sample Means Population 1 Population 2

19. Sampling Distribution of the Difference between Two Sample Means

20. Z Formula for the Difference in Two Sample Means When  1 2 and  2 2 are known and the two samples are independent Samples

21. Hypothesis Testing for Differences Between Means: The Salary Example Advertising Managers 74.25657.79171.115 96.23465.14567.574 89.80796.76759.621 93.26177.24262.483 103.03067.05669.319 74.19564.27635.394 75.93274.19486.741 80.74265.36057.351 39.67273.904 45.65254.270 93.08359.045 63.38468.508 Auditing Managers 69.96277.13643.649 55.05266.03563.369 57.82854.33559.676 63.36242.49454.449 37.19483.84946.394 99.19867.16071.804 61.25437.38672.401 73.06559.50556.470 48.03672.79067.814 60.05371.35171.492 66.35958.653 61.26163.508 Random sample of 32 advertising and 34 auditing managers was taken. The sample statistics are given below. Is there a difference between the sample means?

22. Hypothesis Testing for Differences Between Means: The Salary Example  =0.05,  /2 = 0.025, z 0.025 =  1.96 Hypothesis: Are the salaries in the two functions different. The better way to perform a hypothesis for two populations is to use the difference of the means as the null hypothesis.

23. Hypothesis Testing for Differences Between Means: The Salary Example Since the observed value of 2.35 is greater than 1.96, reject the null hypothesis. That is, there is a significant difference between the average annual wage of advertising managers and the average annual wage of an auditing manager.

24. Exercise: Hypothesis Testing for Differences Between Means (Two sided) Two independent samples are to be compared. Use Alpa = 0.1% (0.001);

25. Exercise: Hypothesis Testing for Differences Between Means (One sided) Use Alpa = 1% (0.001); z critical = +/- ????

26. Confidence Interval to Estimate  1 -  2 When  1,  2 are known By how much are the means different??

27. Demonstration Problem 10.2

28. EXCEL Output for Hernandez New-Employee Training Problem t-Test: Two-Sample Assuming Equal Variances Variable 1Variable 2 Mean 4 7.7356.5 Variance 19.495 18.27 Observations1512 Pooled Variance 18.957 Hypothesized Mean Difference0 df25 t Stat - 5.20 P(T<=t) one-tail1.12E-05 t Critical one-tail1.71 P(T<=t) two-tail2.23E-05 t Critical two-tail 2.06

29. Dependent Samples Before and after measurements on the same individual Studies of twins Studies of spouses Individual 12345671234567 Before 32 11 21 17 30 38 14 After 39 15 35 13 41 39 22

30. Formulas for Dependent Samples

31. Sheet Metal Example-EXCEL Solution