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Posted on Jul 05, 2024

2. (12 marks) In a study of hair, $7 university students each provided one hair, The diameter of each hair (in mm) is recorded. along

2. (12 marks) In a study of hair, $7 university students each provided one hair, The diameter of each hair (in mm) is recorded. along with the student's sex (female or male) and hair colour (black. blond, brown, or red). The data are available in the file hair. txt. You may assume that diameters are independent and normally distributed with common SD. a. (2 marks) Write down the factor effects version of a 2-way ANOVA model with main effects only for these data. Be sure to define your variables and to state all assumptions. b, (1 mark) Provide the ANOVA table associated with the fitted model. The first row of your table (after the intercept) should correspond to sex, the second to colour. c. (2 marks) In words, what is the null hypothesis of the test associated with the p-value in the first row (after the intercept) of your ANOVA table? d. (2 marks) In words; what is the null hypothesis of the test associated with the p-value in the second row (after the intercept) of your ANOVA table? e. (5 marks) Conduct the text of whether hair diameter depends on both sex and colour, In particular i, Specify the null and alternative hypotheses. H. Specify the value of the test statistic. in. Specify the p-value. iv. Using a significance level of a = 0.05, state your conclusions in the language of the problem.3. (15 marks) A tea company wants to assess the relationship between amount of caffeine in their black tea and the brewing time. They take a random sample of leaves from 30 batches of black tea. They make one, 8-of cup of tea from each sample (30 cups altogether), brewing cach for a time ranging from 1 to 10 minutes. They then measure the amount of caffeine (in my) in each, You may assume that the caffeine measurements are independent. The data are available in the file blacktea . txt. a. (2 marks) Produce a scatterplot to illustrate the relationship between caffeine and brewing time. Be sure to label the axes appropriately. b. (2 marks) Write down the linear regression model for these data, Be sure to define your variables and to state all assumptions. c. (2 marks) Provide an interpretation of the effect of brewing time in your model in b). d. (I mark) Fit the model in (b) and provide the table of parameter estimates, standard errors, etc. c. (2 marks) What is the estimated mean amount of caffeine for a brewing time of 2 minutes? f. (2 marks) Are the caffeine measurements approximately normally distributed? Justify your answer. g (2 marks) Is the standard deviation of the caffeine measurements approximately the same for each brewing time? Justify your answer. h. (2 marks) Provide a 95% confidence interval for the effect of brewing time in your model in b)."Caffeine" "BrewTime" 48 3 100 10 71 8 01 10 80 8 68 2 67 6 64 6 101 7 100 10 45 1 30 82 8 100 10 39 3 104 9 67 4 109 10 90 7 49 2 110 4 79 8 72 6av 23 64 57 59 75 68 57 34 63 55 82 99 73 99 99 63 80 51 90 82 68 73 63 54 nun n "Blond\" \"Blond" "Blond" II Blond II II "Blond" "Blond" \" \"Blond" \"Blond\" \"Blond" \"Blond\" \"Brown\" \"Brown" \"Brown\" \"Brown\" "Brown" \"Brown\" \"Brown" \"Brown" \"Brown" \"Brown" \"Brown\" \"Brown\" \"Brown\" "Brown" \"H" \"F" \"all 101 "Black\" \"M\" 103 140 111 132 141 144 111 120 113 125 11? 119 124 13? 152 119 125 122 123 109 136 154 165 122 123 109 125 98 113 "Black\" "Black\" "Black\" "Black\" "Black\" \"Black\" "Black\" \"Black\" "Black\" "Black\" \"Black\" \"Black\" "Black" \"Black\" "Black\" \"Black\" \"Black\" "Black\" "Black\" \"Black\" \"Black\" \"Black" \"Black\" \"Black" \"Black\" \"Black\" \"Black\" \"Black" "Black\" \"M\" IIMII \"F" \"all \"F" \"Diameter" \"Colour" \"Sex" 76 \"Red" "H" 81 \"Red" "H" 40 "Red\" "H" 23 \"Blond" "H" 64 "Blond" "F" 5? \"Blond" "H" 59 "Blond" "F" 75 \"Blond" "F" 68 \"Blond" "F" 5? \"Blond" "H" 34 \"Blond" "H" 63 "Blond" "H" 55 \"Blond" "H" 82 "Brown" \"F" 99 "Brown" "M" 73 \"Brown" "M" 99 \"Brown" "M" 99 \"Brown" "F" 63 \"Brown" "M" 80 \"Brown" "M\" 51 \"Brown" "M" 90 \"Brown" "M 82 "Brown" M 68 "Brown\" "H\" 73 "Brown" \"M 63 \"Brown" M 54 \"Brown" \"F" 101 \"Black\" "M\" 103 "Black\" "M" 140 \"Black" \"M\" 111 \"Black\" \"F\" 132 \"Black" \"M\" 141 "Black" "F" 144 \"Black\" "F" 111 "Black\" "F" 120 "Black" "M" 113 "Black\" "F" 125 "Black\" "M" 117 "Black\" "M" 119 \"Black" "F\" 124 "Black" "M" 13? \"Black" "F" 152 \"Black" "M" 119 "Black" "M" 125 "Black" "M" 122 "Black" "F" 123 "Black" \"F" 109 "Black" \"F" 136 \"Black" \"F" 154 "Black" \"M" 165 "Black\" "M\" 111 "n1anv" "u"