1. Bennie D Waller, Longwood University Hypothesis testing Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901

2. Bennie D Waller, Longwood University  HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing.  HYPOTHESIS TESTING A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement. 10-2 Hypothesis Testing

3. Bennie D Waller, Longwood University Important Things to Remember about H 0 and H 1 H 0 is always presumed to be true H 1 has the burden of proof A random sample (n) is used to “reject H 0 ” If we conclude 'do not reject H 0 ', this does not necessarily mean that the null hypothesis is true, it only suggests that there is not sufficient evidence to reject H 0 ; rejecting the null hypothesis then, suggests that the alternative hypothesis may be true. Equality is always part of H 0 (e.g. “=”, “≥”, “≤”). “≠” “ ” always part of H 1 10-3 Hypothesis Testing

4. Bennie D Waller, Longwood University 10-4 Hypothesis Testing Hypothesis Setups for Testing a Mean (  ) Tests Concerning Proportion

5. Bennie D Waller, Longwood University One-tail vs. Two-tail Test 10-5 Hypothesis Testing

6. Bennie D Waller, Longwood University Hypothesis Testing Dominos Mean32 Variance40 N35 Std. error1.07 T-value1.87 H 0 : µ D = 30 H 1 : µ D ≠ 30 @.10 level Z=1.645 @.05 level Z=1.96 @.01 level Z=2.33

7. Bennie D Waller, Longwood University Hypothesis Testing Dominos Mean32 Variance40 N35 Std. error1.07 T-value1.87 @.05 level Z=1.645H 0 : µ D ≤ 30 H 1 : µ D > 30

8. Bennie D Waller, Longwood University 10-8 Hypothesis Testing

9. Bennie D Waller, Longwood University Hypothesis Testing Problem: The waiting time for patients at local walk-in health clinic follows a normal distribution with a mean of 15 minutes and a population standard deviation of 5 minutes. The quality- assurance department found in a sample of 50 patients that the mean waiting time was 14.25 minutes. At the 0.025 significance level, decide if the sample data support the claim that the mean waiting time is less than 15 minutes. State your decision in terms of the null hypothesis. z0.000.010.020.030.040.050.060.070.080.09 1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319 1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441 1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545 1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633 1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706 1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767 2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817

10. Bennie D Waller, Longwood University Hypothesis Testing Problem: A manufacturer wants to increase the shelf life of a line of cake mixes. Past records indicate that the average shelf life of the mix is 216 days. After a revised mix has been developed, a sample of nine boxes of cake mix had a mean of 217.222 and a standard deviation of 1.2019. At the 0.025 significance level, decide if the sample data support the claim that shelf life has increased. State your decision in terms of the null hypothesis.

11. Bennie D Waller, Longwood University Hypothesis testing – Two samples Bennie Waller wallerbd@longwood.edu 434-395-2046 Longwood University 201 High Street Farmville, VA 23901

12. Bennie D Waller, Longwood University Two-Sample Hypothesis Testing H 0 : µ PJ = µ D H 1 : µ PJ ≠ µ D H 0 : µ PJ - µ D = 0 H 1 : µ PJ - µ D ≠ 0 Setting up a hypothesis test to see if there is a difference between the average delivery time of two pizza delivery companies. H 0 : µ PJ - µ D = 5 H 1 : µ PJ - µ D ≠ 5 Can test for difference in any value. Typically test for zero.

13. Bennie D Waller, Longwood University Two-Sample Hypothesis Testing H 0 : µ PJ ≤ µ D H 1 : µ PJ > µ D H 0 : µ D ≥ µ PJ H 1 : µ D < µ PJ H 0 : µ PJ - µ D ≤ 0 H 1 : µ PJ - µ D > 0 Setting up a hypothesis test to see if there is a difference between the average delivery time of two pizza delivery companies.

14. Bennie D Waller, Longwood University Two-Sample Hypothesis Testing Comparing Two Population Means - Example Step 1: State the null and alternate hypotheses. H 0 : µ PJ ≤ µ D H 1 : µ PJ > µ D Step 2: Select the level of significance. For example a.01 significance level. Step 3: Determine the appropriate test statistic. If the population standard deviations are known, use z-distribution as the test statistic, otherwise use t-statistic.

15. Bennie D Waller, Longwood University Step 4: Formulate a decision rule. Reject H 0 ifZ > Z  Z > 2.33 11-15 Two-Sample Hypothesis Testing

16. Bennie D Waller, Longwood University Comparing Two Population Means: Equal Variances No assumptions about the shape of the populations are required. The samples are from independent populations. The formula for computing the value of z is: 11-16 Two-Sample Hypothesis Testing

17. Bennie D Waller, Longwood University Two-Sample Hypothesis Testing DominosPapa Johns Mean3538 Variance6048 N3540 Variance/N1.711.2 Std. error1.707 T-value-1.76 H 0 : µ PJ - µ D = 0 H 1 : µ PJ - µ D ≠ 0 @.10 level Z=1.645 @.05 level Z=1.96 @.01 level Z=2.33

18. Bennie D Waller, Longwood University Two-Sample Hypothesis Testing DominosPapa Johns Mean3538 Variance6048 N3540 Variance/N1.711.2 Std. error1.707 T-value-1.76 H 0 : µ D ≥ µ PJ H 1 : µ D < µ PJ @.05 level Z=1.645 @.01 level Z=2.33

19. Bennie D Waller, Longwood University Two-Sample Hypothesis Testing

20. Bennie D Waller, Longwood University Comparing Population Means with Equal but Unknown Population Standard Deviations (the Pooled t-test) The t distribution is used as the test statistic if one or more of the samples have less than 30 observations. The required assumptions are: 1.Both populations must follow the normal distribution. 2.The populations must have equal standard deviations. 3.The samples are from independent populations. 11-20 Two-Sample Hypothesis Testing

21. Bennie D Waller, Longwood University Finding the value of the test statistic requires two steps. 1.Pool the sample standard deviations. 2.Use the pooled standard deviation in the formula. 11-21 Two-Sample Hypothesis Testing

22. Bennie D Waller, Longwood University Comparing Population Means with Unknown Population Standard Deviations (the Pooled t-test) - Example Step 1: State the null and alternate hypotheses. (Keyword: “Is there a difference”) H 0 : µ 1 = µ 2 H 1 : µ 1 ≠ µ 2 Step 2: State the level of significance. The 0.10 significance level is stated in the problem. Step 3: Find the appropriate test statistic. Because the population standard deviations are not known but are assumed to be equal, we use the pooled t-test. 11-22 Two-Sample Hypothesis Testing

23. Bennie D Waller, Longwood University Comparing Population Means with Unknown Population Standard Deviations (the Pooled t-test) - Example Step 4: State the decision rule. Reject H 0 ift > t  /2,n 1 +n 2 -2 or t < - t  /2, n 1 +n 2 -2 t > t.05,9 or t < - t.05,9 t > 1.833 or t < - 1.833 11-23 Two-Sample Hypothesis Testing

24. Bennie D Waller, Longwood University Problem: A financial planner wants to compare the yield of income and growth mutual funds. Fifty thousand dollars is invested in each of a sample of 35 income and 40 growth funds. The mean increase for a two-year period for the income funds is $900. For the growth funds the mean increase is $875. Income funds have a sample standard deviation of $35; growth funds have a sample standard deviation of $45. Assume that the population standard deviations are equal. At the 0.05 significance level, is there a difference in the mean yields of the two funds? What decision is made about the null hypothesis using alpha = 0.05? Two-Sample Hypothesis Testing

25. Bennie D Waller, Longwood University End