STA 412 HW 06 (Contrasts F-test) Question: 1 Total Points: 30 30 Bonus Points: Score: 0 0 1. (Textbook, Exercise 9.13)Researchers conducted an experiment to compare the effectiveness of four new weight- reducing agents to that of an existing agent. The researchers randomly divided a random sample of 50 subjects into five equal groups, with preparation Al assigned to the first group, A2 to the second group, and so on. They then gave a prestudy physical to each person in the experiment and told them how many pounds overweight they were. A comparison of the mean number of pounds overweight for the groups showed no significant dif- ferences. The researchers then began the study program, and each group took the prescribed preparation for a fixed period of time. The four new agents are labeled A1, A2, A3, and A4, and the standard agent is labeled agent S, as shown in the table below. Subscript Agent Population mean Sample size 1 A1 με 10 2 A2 μ2 10 3 A3 из 10 4 A4 μ4 10 5 S με 10 The weight losses recorded at the end of the study period are given in the file "hw5_data.txt". You can read the data into R using the following code. df <- read.table("hw06_data.txt", header = TRUE, strings AsFactors = TRUE) df$agent # agent is a factor ## [1] A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A3 A3 A3 A3 A3 A3 A3 A3 A3 A3 A4 A4 ## [33] A4 A4 A4 A4 A4 A4 A4 A4 S S S S S S SS S S ##23 Levels: A1 A2 A3 A4 S (a) (5 points) Verify that the contrasts below are mutually orthogonal. 1 = μ1 + μ2 + μ3 + μ4 - 4μ5 2 = μ1 + μ2 - μ3 - μ4 3 = 1 - 2 + μ3 - μ4 l₁ = μ1 - μ2 - μ3 + μ4 = = μ5 against Ha at least one of (b) (5 points) Consider the hypothesis testing problem Ho: μ₁ μ1,..., μ5 is different. Use R to generate an ANOVA table. (c) (10 points) Use the glht function in R to construct 95% two-sided confidence intervals for each contrast in (a) (without correction for multiple comparisons). (d) (10 points) Use the aov function in R to generate an ANOVA table including the sum of squares, mean square, F test statistic for each contrast in (a). Based on the ANOVA table, what are the contrasts that are statistically significantly different from zero (a 0.05, without correction for multiple comparisons)? =