Stat 512 - Exam 2Print out this exam and provide handwritten answers in the space provided. Bringthe answers to class on Monday.1. A study was conducted to measure the effects of heredity and environment onintelligence. Children were selected at random from an adoption registry inFrance in four categories determined by the socioeconomic status (SES, highor low) of their adoptive parents and their biological parents. Each child wasgiven an IQ test. The group means are shown below.Biological parentsAdoptive parents High SES Low SESHigh SES 120 104Low SES 108 92Suppose that there are n = 5 subjects per group, and suppose that the errorsum of squares is SSErr= 3200.(a) Write the linear model for the two-way factorial analysis, and find theleast-squares estimates for all terms in the model (except the ²’s).(b) Construct the ANOVA table. Which effects (main effects, interaction)are statistically significant?2. Researchers conducted a clinical trial to assess the effectiveness of an over-the-counter allergy medication in reducing allergy symptoms. The drug wasdelivered in two forms (pill or liquid) at three dosage levels (low, medium,high) in a completely randomized 2 × 3 design with n = 10 subjects percell. The response variable is the severity of allergy symptoms. The averageresponses in each cell are shown below.Factor B: DoseFactor A: Form Low Medium HighPill 6.5 5.9 5.9Liquid 6.6 5.9 5.2Suppose that S2= MSErr= 1.0.(a) Find SSReg, the regression sum of squares, for the linear mo del that in-cludes the main effect for A, the main effect for B, and the AB interaction.How many degrees of freedom does it have?(b) Decompose SSRegin the usual way. That is, find SSA, SSBand SSAB.Summarize the results with an ANOVA table, and report the F-statistics.(c) Decompose SSBinto independent pieces corresponding to the linear andquadratic effects of dose. (Use the pretty integer weights −1, 0, +1 forthe linear effect and +1, −2, +1 for the quadratic effect.) Are either ofthese effects significant?(d) Define contrasts to testL1: the linear effect of dose when A=pillL2: the linear effect of dose when A=liquidAre either of these contrasts significant? Should these be considered“interaction contrasts” (i.e., do they represent portions of SSAB)? Howdo you know?(e) Now define contrasts to testL3: the quadratic effect of dose when A=pillL4: the quadratic effect of dose when A=liquidAre either of these contrasts significant? Are they “interaction con-trasts”?(f) Add up the sums of squares for the four contrasts (L1, L2, L3, L4) fromparts (d) and (e). To what does this sum correspond in the ANOVAtable from part