Exam 13: Comparing Three or More Means

A researcher wishes to determine whether there is a difference in the average age of middle school, highschool, and college teachers. Teachers are randomly selected. Their ages are recorded below. Test thehypothesis that there is no difference in the average age of each group. Use α = 0.01. Middle School Teachers High School Teachers College Teachers 23 41 39 28 38 45 27 36 45 52 47 61 25 42 36 37 31 35

Which of the following is not an example of a mean square?

Assume that the data below come from populations that are normally distributed with the same variance. Let $\mu _ { 1 }$ be the mean for treatment $1 , \mu _ { 2 }$ the mean for treatment 2 , and $\mu _ { 3 }$ the mean for treatment 3 . In a test of $H _ { 0 }$ : $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 }$ versus $\mathrm { H } _ { 1 }$ : at least one of the means is different, the null hypothesis is rejected at the $\alpha = 0.05$ level of significance. Use Tukey's test to determine which pairwise means differ using a familywise error rate of $\alpha = 0.05$ . Give the P-value for each of the pairwise tests and state your conclusion. Block Treatment 1 Treatment 2 Treatment 3 1 17.3 19.1 16.9 2 17.9 21.4 17.5 3 17.2 23.3 18.6 4 19.1 21.6 20.1 5 20.8 23.7 19.6

Four different types of insecticides are used on strawberry plants. The number of strawberries on eachrandomly selected plant is given below. Find the critical value F-value to test the hypothesis that the type ofinsecticide makes no difference in the mean number of strawberries per plant. Use α = 0.01. Insecticide 1 Insecticide 2 Insecticide 3 Insecticide 4 6 8 6 3 6 5 3 5 5 5 4 3 7 5 2 4 7 5 3 5 6 6 3 4

Find the critical value from the Studentized range distribution for α = 0.01, ν = 10, k = 4.

A two-way analysis of variance is to be performed using the data below. Assume that the data come frompopulations that are normally distributed with the same variance. If there is a significant difference in themeans for the three levels of factor A, use Tukeyʹs test to determine which pairwise means differ using afamilywise error rate of α = 0.05. If there is a significant difference in the means for the three levels of factor B,use Tukeyʹs test to determine which pairwise means differ using a familywise error rate of α = 0.05. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Factor B }$ Level 1 Level 2 Level 3 Level 1 67 74 82 71 73 76 62 60 87 Level 2 77 69 84 Factor A 81 75 80 65 77 90 Level 3 59 70 72 67 58 68 62 66 76

True or False: It is required for the Two-Way ANOVA that the samples be dependent.

A certain HMO is attempting to show the benefits of managed care to an insurance company. The HMObelieves that certain types of doctors are more cost-effective than others. One theory is that Certification Levelis an important factor in measuring the cost-effectiveness of physicians. To investigate this, the HMO obtainedindependent random samples of 19 physicians from each of the three certification levels-- Board certified (C);Uncertified, board eligible (E); and Uncertified, board ineligible (I)-- and recorded the total per-member,per-month charges for each (a total of 19 physicians). In order to compare the mean charges for the threegroups, the data will be subjected to an analysis of variance.Write the null hypothesis tested by the ANOVA.

A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with highcholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is givenmedication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of sixweeks, each subjectʹs cholesterol level is recorded. State the requirements that must be satisfied in order to usethe one-way ANOVA procedure.

Assume that the data below come from populations that are normally distributed with the same variance. Let $\mu _ { 1 }$ be the mean for treatment $1 , \mu _ { 2 }$ the mean for treatment 2 , and $\mu _ { 3 }$ the mean for treatment 3 . In a test of $\mathrm { H } _ { 0 }$ : $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 }$ versus $\mathrm { H } _ { 1 }$ : at least one of the means is different, the null hypothesis is rejected at the $\alpha = 0.05$ level of significance. Use Tukey's test to determine which pairwise means differ using a familywise error rate of $\alpha = 0.05$ . Give the $\mathrm { P }$ -value for each of the pairwise tests and state your conclusion. Block Treatment 1 Treatment 2 Treatment 2 1 7.2 6.6 6.1 2 6.9 7.5 6.7 3 8.2 8.6 8.3 4 9.7 9.3 8.8 5 9.4 10.2 9.5 6 10.3 9.8 9.2

A farmer has five different fields. He wishes to test two different fertilizers. The yields of corn may vary from field to field depending on a number of factors such as soil quality, sunlight etc. He divides each field into three sections. For each field, he randomly assigns one of the sections to receive fertilizer A, one of the sections to receive fertilizer B, and one of the sections to receive no fertilizer. At harvest time, he records the corn yields in bushels per acre for each section of each field. The results are shown in the table. Let $\mu 1$ be the mean for fertilizer $\mathrm { A } , \mu _ { 2 }$ the mean for fertilizer $\mathrm { B }$ , and $\mu _ { 3 }$ the mean for no fertilizer. In a test of $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 }$ versus $\mathrm { H } _ { 1 }$ : at least one of the means is different, the null hypothesis is rejected at the $\alpha = 0.05$ level of significance. Use Tukey's test to determine which pairwise means differ using a familywise error rate of $\alpha = 0.05$ . Give the P-value for each of the pairwise tests and state your conclusion. Fertilizer A Fertilizer B No Fertilizer Field 1 132 141 128 Field 2 136 135 130 Field 3 144 147 139 Field 4 126 134 122 Field 5 130 128 125

Test the hypothesis that the populations have the same mean. Use α= 0.05. Brand 1 Brand 2 Brand 3 =8 =8 =8 1=3.0 2=2.6 3=2.6 =0.50 =0.60 3=0.55

A researcher is testing the percent weight gain on mice of varying ages based upon the diet they eat. He has 4mice in each age category: 6 months, 9 months, 12 months and 15 months. They mice are weighed, fed onlyone of four diets and weighed after 1 month. The weight gain percent is then recorded. The data is givenbelow. Test at α = 0.05 if there is a difference in weights. Diet A Diet B Diet C Diet D 6 months 5.2 5.3 5.8 5.6 9 months 5.0 5.7 5.6 5.2 12 months 5.3 6.1 6.1 5.0 15 months 4.9 6.4 5.7 5.1

A researcher wishes to determine whether there is a difference in the average age of middle school, highschool, and college teachers. Teachers are randomly selected. Their ages are recorded below. Find the criticalvalue F-value to test the hypothesis that there is no difference in the average age of each group. Useα = 0.01. Middle School Teachers High School Teachers College Teachers 23 38 39 28 36 45 27 41 36 52 47 61 25 42 45 37 31 35

The analysis of variance is an inferential method that is used to test the equality of

To conduct an analysis of variance on a randomized complete block design a researcher would use

A farmer has a 160 acre tract of land that he is planting in corn. He has heard about advances in fertilizertechnology that impacts yield. He divides his field into four sections and each section into four parts. He hasthree type of fertilizer to test and no fertilizer. He randomly selects a part of a section and then randomlyselects a type of fertilizer for that part of a section so that each section has all four applications applied to it.The yields in bushel per acres are given below. Test at α = 0.05 if there is a difference in yields. Field A Field B Field C Field D No fertilizer 5.7 10.4 7.3 9.0 Traditional 4.5 8.6 6.1 10.6 Liquid 4.1 6.3 5.8 7.2 Granular 2.6 7.2 4.6 8.7

A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with highcholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is givenmedication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of sixweeks, each subjectʹs cholesterol level is recorded. Use α = 0.05. Which of the following is a correct statementabout the means? Group 1 Group 2 Group 3 13 8 6 12 5 12 9 2 4 15 3 4 11 4 9 8 0 8