Section 10-4 – Analysis of Variance One-Way ANOVA One-way analysis of variance (ANOVA) is a hypothesis-testing technique that is used to compare means from three or more populations. To begin an ANOVA test, you should first state a null and an alternate hypothesis. For a one-way ANOVA test, the null and alternate hypotheses are always similar to the following statements: H0: µ1 = µ2 = µ3 = ….. µk. (All population means are equal) Ha: At least one of the means is different from the others. To reject the H0 means that at least one of the means is different from the others. It will take more testing to determine which one of the means is different from the others.

Section 10-4 – Analysis of Variance One-Way ANOVA In an one-way ANOVA test, the following conditions must be true. 1) Each sample must be randomly selected from a normal, or approximately normal, population. 2) The samples must be independent of each other. 3) Each population must have the same variance. Guidelines For Finding the Test Statistic for a One-Way ANOVA Test. This test can be run on the TI-84 by entering each set of data into STAT-EDIT using L1, L2, L3, etc. Once the data has been entered, go to STAT-TEST-H (ANOVA). Enter the names of the lists into which you put the data, separated by commas, and allow the calculator to do the work.

Section 10-4 – Analysis of Variance One-Way ANOVA Example 1 – Page 591 A medical researcher wants to determine whether there is a difference in the mean length of time it takes three types of pain relievers to provide relief from headache pain. Several headache sufferers are randomly selected and given one of the three medications. Each headache sufferer records the time (in minutes) it takes the medication to begin working. The results are shown in the table. At α = 0.01, can you conclude that the mean times are different? Assume that each population of relief times is normally distributed and that the population variances are equal. H0: µ1 = µ2 = µ3 Ha: At least one mean is different from the others. (claim) Med 1 Med 2 Med 3 12 16 14 15 17 21 20 19

Section 10-4 – Analysis of Variance One-Way ANOVA Example 1 – Page 591 Use the calculator to enter these lists into STAT-EDIT. Med 1 into L1 Med 2 into L2 Med 3 into L3 Once the lists are entered, go to STAT-TEST-H (ANOVA) Press “2nd 1, 2nd 2, 2nd 3” and close the parenthesis. You must use the commas!! Comma key is above the 7 key. Press Enter. Med 1 Med 2 Med 3 12 16 14 15 17 21 20 19

Section 10-4 – Analysis of Variance One-Way ANOVA Example 1 – Page 591 The calculator gives us many values and numbers. For our purposes, we are going to look at the p-value and compare it to α to make our decision. Since p = .269, which is greater than 0.01, we will fail to reject the null. Because we failed to reject the null, we can not support the claim. There is not enough evidence at the 1% significance level to conclude that there is a difference in the mean length of time it takes the three pain relievers to provide relief from headache pain.

Section 10-4 – Analysis of Variance One-Way ANOVA Example 2 – Page 593 Three airline companies offer flights between Corydon and Lincolnville. Several randomly selected flight times (in minutes) between the towns for each airline are shown. Assume that the populations of flight times are normally distributed, the samples are independent, and the population variances are equal. At α = 0.01, can you conclude that there is a difference in the means of flight times? H0: µ1 = µ2 = µ3 Ha: At least one mean is different from the others. Airline 1 Airline 2 Airline 3 122 119 120 135 133 158 126 143 155 131 149 125 114 147 116 124 164 134 108 151 142 140 113 136 141

Section 10-4 – Analysis of Variance One-Way ANOVA Example 2 – Page 593 Use the calculator to enter these lists into STAT-EDIT. Airline 1 into L1 Airline 2 into L2 Airline 3 into L3 Once the lists are entered, go to STAT-TEST-H (ANOVA) Press “2nd 1, 2nd 2, 2nd 3” and close the parenthesis. You must use the commas!! Press Enter. Airline 1 Airline 2 Airline 3 122 119 120 135 133 158 126 143 155 131 149 125 114 147 116 124 164 134 108 151 142 140 113 136 141

Section 10-4 – Analysis of Variance One-Way ANOVA Example 2 – Page 593 You can see from the calculator that the p-value of this test is 0.0064. Since this is less than the α value of 0.01, we need to reject the null hypothesis. At least one of the flight times has a mean different from the others. There is enough evidence at the 1% significance level to conclude that at least one of the flight times has a different mean from the others.

Assignments: Classwork: Pages 595-599 #1-2, 5–15 Odds Homework: Pages 595-599 #6–16 Evens For #5–16 part b), don’t worry about the rejection region; we use p instead.