Stat 512 - Assignment 3Due Monday, March 291. Adapted from Kuehl, Exercise 6.3. A study was conducted to assess the effectof temperature on the shrinkage of dyed fabric. Four temperatures were ap-plied to four different fabrics in a 4 × 4 crossed design with n = 2 replicationsper cell. The response variable is the percent shrinkage. The data are shownbelow.TemperatureFabric 210◦F 215◦F 220◦F 225◦F1 1.8, 2.1 2.0, 2.1 4.6, 5.0 7.5, 7.92 2.2, 2.4 4.2, 4.0 5.4, 5.6 9.8, 9.23 2.8, 3.2 4.4, 4.8 8.7, 8.4 13.2, 13.04 3.2, 3.6 3.3, 3.5 5.7, 5.8 10.9, 11.1Data for this problem are available in the file FabricShrink.txt. The datacan be entered into R by typing> FabricShrink <- read.table("FabricShrink.txt", header=T)at the R command prompt. Then be sure to change the variables fabric andtemp into factors.(a) Plot the response variable y versus each factor, and comment on whatyou see.(b) Fit the two-way ANOVA model using y as the response. Are the effectsof the two factors additive?(c) Compute the mean for each level of temp (the¯Y·j·’s) and plot them.Decompose the SS for main effect of temp into independent SS’s corre-spnding to linear, quadratic and cubic contrasts. Which of these trendsare significant?(d) Compute the cell means (the¯Yij·’s) and plot them to show the effect oftemperature on shrinkage for each fabric.(e) Compute the mean for each fabric (the¯Yi··’s). Which fabric appears toshrink the most, and which fabric appears to shrink the least? Definea main-effect contrast to test whether the µi·’s for these two fabrics aresignificantly different. Apply a Bonferroni adjustment, treating this asthe largest ofÃ42!= 6pairwise differences.2. The dataset below, reported by Box and Cox (1964, JRSS-B), shows theresults from a 3 × 4 completely randomized, balanced factorial experiment.Animals were exp osed to three different poisons and given four different trest-ments. The response is survival time.TreatmentPoison A B C DI 0.31 0.82 0.43 0.450.45 1.10 0.45 0.710.46 0.88 0.63 0.660.43 0.72 0.76 0.62II 0.36 0.92 0.44 0.560.29 0.61 0.35 1.020.40 0.49 0.31 0.710.23 1.24 0.40 0.38III 0.22 0.30 0.23 0.300.21 0.37 0.25 0.360.18 0.38 0.24 0.310.23 0.29 0.22 0.33Data for this problem are available in the file PoisonAnimals.txt. The datacan be entered into R by typing> PoisonAnimals <- read.table("PoisonAnimals.txt", header=T)at the R command prompt.(a) Plot the response, survival time, versus each factor and comment on whatyou see.(b) Fit the two-way ANOVA model to survival time and test for additivity.Plot the residuals versus the fitted values. Comment.(c) Plot the log(survival time) versus each factor and comment.(d) Fit the two-way ANOVA model using log(survival time) as the response,and test for additivity. Plot the residuals versus the fitted values. Com-ment.(e) Which treatment seems to give the longest survival time? The shortest?Is the difference between these two treatments statistically significant?Perform this test using the model for log(survival time) that you fit inpart (b), and include a Bonferroni adjustment that treats this as thelargest of six pairwise differences.3. Adapted from KNNL, Exercise 19.12. Psychologists conducted an experimentto investigate the effect of eye contact on personnel officers’ assessments ofthe likely success of job candidates. Twenty personnel officers (ten male, tenfemale) were shown front-view photographs of an applicant’s face and wereasked to rate the potential for success on a scale from 0 (total failure) to 20(outstanding success). Half of the officers received a photograph in which theapplicant made direct eye contact with the camera lens; the other half receiveda photograph without direct eye contact. The responses are shown below.SexEye contact M FPresent 11 157 1212 146 1110 16Absent 12 1416 1710 1313 2014 18Data for this problem are available in the file EyeContact.txt. The data canbe entered into R by typing> EyeContact <- read.table("EyeContact.txt", header=T)at the R command prompt.(a) Fit the two-way ANOVA model. Is there an interaction between the twofactors (sex and eye contact)? Do you think it is sensible to eliminatethe interaction from this model? Do you think there is any advantage todoing so?(b) Compute a 95% confidence interval for (i) the effect of eye contact amongmales, µ11−µ21, and (ii) the effect of eye contact among females, µ12−µ22,under the model fit in part (a) which allows for an interaction.(c) Fit the model without an interaction and compute a 95% confidenceinterval for the effect of eye contact (yes versus no) in the combined pop-ulation of males and females. One easy way to do this is to apply dummycoding and examine the results from the table of coefficients. Comparethe width of this confidence interval to the widths of the intervals youfound in part (b), and