Discuss b conduct separate tests for type of group

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Discuss. b. Conduct separate tests for type of group and size of group main e ff ects. In each test, use level of significance α = . 01 and state the alternatives, decision rule, and conclusion. What is the P -value for each test? c. Obtain confidence intervals for D 1 = μ . 2 - μ . 1 , D 2 = μ . 3 - μ . 2 , and D 3 = μ . 4 - μ . 3 ; use the Bonferroni procedure with a 95 percent family confidence coe ffi cient. State your findings. d. Is the Bonferroni procedure used in pan (c) the most e ffi cient one here? Explain. Solution: a. The plot of the observations are shown below: 4
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It appears that interaction e ff ects are very little, since the two curves are nearly parallel. Factor A main e ff ects are also very little, since the two curves are nearly overlapped. Factor B main e ff ects are present, since the two curves have large departure from horizontal. b. We first construct the ANOVA table. Note that the error sum of squares is given by S S AB , which can be obtained by S S AB = S S TO - S S A - S S B in R. Source S S d f MS Type of group 1.125 1 1.125 Size of group 318.375 3 106.125 Error 6.375 3 2.125 Total 325.875 7 F test for type of group main e ff ects: H 0 : α 1 = α 2 = 0, H a : not both α 1 and α 2 are 0. F * = MS A / MS AB = 1 . 125 / 2 . 125 = . 53, F (1 - α ; a - 1 , ( a - 1)( b - 1)) = F ( . 99; 1 , 3) = 34 . 1. If F * < 34 . 1, conclude H 0 , otherwise H a . Here conclude H 0 . P -value = .52. H 0 : β 1 = β 2 = β 3 = β 4 = 0, H a : not all β i ’s are 0. F * = MS B / MS AB = 106 . 125 / 2 . 125 = 49 . 94, F (1 - α ; b - 1 , ( a - 1)( b - 1)) = F ( . 99; 3 , 3) = 29 . 5. If F * < 29 . 5, conclude H 0 , otherwise H a . Here conclude H a . P -value = .005. 5
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c. ˆ D 1 = ¯ Y . 2 - ¯ Y . 1 = 6, ˆ D 2 = ¯ Y 3 . - ¯ Y 2 . = 7 . 5, ˆ D 3 = ¯ Y . 4 - ¯ Y . 3 = 2 . 5. s ( ˆ D 1 ) = s ( ˆ D 2 ) = s ( ˆ D 3 ) = 2 MS AB / a = 1 . 4577. B = t (1 - α/ (2 g ); ( a - 1)( b - 1)) = t ( . 99167; 3) = 4 . 857. Thus, the 95% family-wise CI for D 1 , D 2 and D 3 using the Bonferroni procedure are 6 ± 4 . 857(1 . 4577) = [ - 1 . 08 , 13 . 08], 7 . 5 ± 4 . 857(1 . 4577) = [ . 42 , 14 . 58] and 2 . 5 ± 4 . 857(1 . 4577) = [ - 4 . 58 , 9 . 58]. d. Sche ff e’s procedure and Tukey’s procedure are also applicable in part (c). S = ( b - 1) F (1 - α ; b - 1 , ( a - 1)( b - 1)) = 3 F ( . 95; 3 , 3) = 5 . 275, T = 1 2 q (1 - α ; b , ( a - 1)( b - 1)) = 1 2 q ( . 95; 4 , 3) = 4 . 822. Since T < B < S , the Tukey’s procedure is the most e ffi cient in part (c). 21.7 Fat in diets. A researcher studied the e ff ects of three experimental diets with varying fat contents on the total lipid (fat) level in plasma. Total lipid level is a widely used predictor of coronary heart disease. Fifteen male subjects who were within 20 percent of their ideal body weight were grouped into five blocks according to age. Within each block, the three experimental diets were randomly assigned to the three subjects. Data on reduction in lipid level (in grams per liter) after the subjects were on the diet for a fixed period of time follow.
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