March 25-28, 2002Statistics 512Midterm 2March 25-28, 2002This is a take home exam. The following rules apply.1. You may use any textbook, including textbooks on the Web.2. You may use any statistical software.3. Anwer the questions on the exam paper in the space provided. Labeled computer output should be attached as needed. You may alsoattach additional pages of computations for any problem, but I will check these only if you direct me to them.4. You may not seek help from any person except Prof. Altman and Lan Wang. This includes assistance from internet sources such as e-mail, bulletin boards and chat rooms.5. Since students will be taking the exam at any time during the week, you may not discuss this exam with anyone until Friday March 29. 6. Failure to comply with items 4 or 5 could lead to reduction in your grade, or disciplinary action.7. When you have completed the exam, slide it under my door. I have read the rules above and agree to comply with them.Signature ________________________________________________Name (printed) ___________________________________________Statistics 512 Midterm 2 Spring 2002Question 1An investigator hypothesized that exposure of plants to unusually high levels of salinity and temperature inhibits their ability to produce dry matter. A completely randomized experiment was conducted to investigate this claim. Factor salinity had three levels (0, 6, and 12 bars) and factor temperature had 3 levels (15, 20 and 250C). This experiment was conducted in a greenhouse with seven pots available for each treatment combination. Three months following plantation, all plants were harvested and the total dry weight of plant material in each pot was measured. The data appear in the table below. The data areavailable from the class Web page.Salinity 0 bars 6 bars 12 barsTemperature 15 20 25 15 20 25 15 20 2522 50 132 37 59 94 28 45 8333 57 133 36 60 110 34 59 7634 52 123 45 57 112 35 46 7321 58 124 48 72 114 27 60 8620 51 134 35 70 92 36 62 8535 59 122 46 74 113 25 61 7536 48 120 34 76 90 37 41 84a. Write the 2-way fixed effects ANOVA model and explain each of the terms in the model. 2Statistics 512 Midterm 2 Spring 2002b. Fit the model and examine the appropriate residual plots. State which plots you have examined, and what you are looking for on these plots. Note any important or unusual features of the plots.Regardless of your answer in b., do not modify the data.3Statistics 512 Midterm 2 Spring 2002c. Test whether or not there is a statistically significant interaction. State your null and alternative hypotheses using the notation in your response to a. Include the appropriate test statistics, p-value and your conclusion.d. Draw the interaction plot and describe the interesting features. (You may need to print the treatment means in order to fully understand this plot.)4Statistics 512 Midterm 2 Spring 2002e. The investigator hypothesized that at the lowest temperature, there would not be a statistically significant salinity effect. Write two orthogonal contrasts of the treatment means that could be used to test this.f. Carry out the test(s) in part e. 5Statistics 512 Midterm 2 Spring 2002g. The investigator chose to fit a 2nd degree model for salinity and temperature. He wishes to determine if a first degree polynomial (including a linear*linear interaction) would be sufficient. Fit the model and determine if a first degree polynomial fits adequately.6Statistics 512 Midterm 2 Spring 2002h. When the paper was sent for review, one of the referees looked at the ANOVA results and concluded that the effect of salinity on the response (dry matter) is affected by temperature. Is she correct? Justify your answer with a test. Explain why this is the appropriate test of this hypothesis.7Statistics 512 Midterm 2 Spring 2002Question 2A manufacturer of industrial textiles is trying to develop a method to produce synthetic spider silk with high strength. There are 5 types of polymer solution. The manufacturer has prepared 4 replicate batches of fiber for solution type and has measured the tensile strength of the fiber from a sample from each batch. The manufacturer wishes to determine if there are solution effects on tensile strength. Fiber diameter is an important determinant of strength. Each of the solutions should be able to create fibers of the same diameter. However, because the industrial process is not perfected, diameter varies significantly from batch to batch. Hence, the manufacturer decides to use fiber diameter as a covariate in the analysis.The data are below and are also on the class Web page.obs rep solution diameter strength1 1 1 117 73.52 2 1 115 70.93 3 1 103 68.54 4 1 115 71.65 1 2 105 69.36 2 2 102 63.97 3 2 109 71.08 4 2 106 70.49 1 3 108 72.910 2 3 108 71.211 3 3 107 70.312 4 3 116 74.313 1 4 103 69.514 2 4 118 78.815 3 4 103 70.916 4 4 119 77.217 1 5 110 64.618 2 5 110 67.719 3 5 103 62.920 4 5 104 65.8a. Write a linear model for the observations of strength, which includes effects for solution (S), diameter (D) and all of the interactions. Make sure that all of the terms in the model are fully defined, including distribution of any random terms, and any constraints on the model parameters. (Assume that the solution and diameter effects are fixed.)8Statistics 512 Midterm 2 Spring 2002b. Fit the model in part a and test for a treatment by covariate interaction. Include null and alternative hypotheses, test statistic, p-value, and your conclusion written as a sentence. (You do not need to check the residual plots.)c. Fit the ANCOVA model with no interaction. Is there a difference in tensile strength among the fibers from the different polymer solutions, after adjusting for diameter. Include null and alternative hypotheses, test statistic, p-value, and your conclusion written as a sentence. 9Statistics 512 Midterm 2 Spring 2002d. Solutions 2 and 3 differ only by the type of catalyst used. Test whether or not whether fibers produced from these solutions have the same mean strength after adjusting for diameter, and give a 95% confidence interval for the difference.10Statistics 512 Midterm 2 Spring 2002e. The experiment was done by two summer students (BJ and MC) who had an internshipat the lab. Each student prepared and measured two of the replicates for each type of solution. Since the fiber diameter is determined by a step in the manufacturing process which is under control of the student, the engineer in charge was