36-402/608 Homework #4 due 10:30AM 2/111. Pygmalion effect (40 points, 5 each)Load the Sleuth’s Pygmalion dataset (case1302.csv), which is the second featuredcase of Chapter 13, and is found on the Sleuth CD. Do the usual EDA to be sure youunderstand the form of the data. Let’s use µCand µPto represent the populationmeans of score for subjects exposed to control vs. “Pygmalion” treatment, andµC1through µC10to represent the population means of score for subjects in the 10companies.(a) Turn in the R summary(aov()) table that allows you to retain the null hy-pothesis of no interaction between company and treatment in their effects onscore.(b) Turn in the R summary(aov()) table that checks just the “block” effect ofcompany. Turn in the null hypothesis of interest (formula or word format,being sure to refer to population not sample means), and state whether or notyou have sufficient evidence to reject it.(c) Turn in the R summary(aov()) that tests for the additional effect of treatmentafter adjusting (correcting) for the effect of company (without interaction).Turn in the null hypothesis of interest and state whether or not you havesufficient evidence to reject it.(d) What is the best estimate of σ2, the within-group error variance, for the modelin part c?(e) What is the p-value for treatment when it is put in the additive model beforecompany?(f) Make a residual plot for the model from part c. Briefly state your conclusionsreached from examining this plot.(g) Run qqn() (from http://www.stat.cmu.edu/∼hseltman/files/qqn.R) on theresiduals from same model as part c, and briefly state your conclusions reachedfrom examining this plot.(h) Why is the p-value from summary(aov(score∼company+treat,pyg)) smallerthan the p-value from summary(aov(score∼treat,pyg)), and how does thisrelate to improving power? (Substitute your data.frame name for “pyg”.)2. Stepping and heart rate (45 points)The study examined in this problem is about the effects of “stepping” exercise onheart rate (HR). Subjects were randomly assigned to two different heights of steps,and three different frequencies. Instructor differences may occur, so instructors weretreated as blocks. (We will not use the “order” or “RestHR” variables.)Load the data from “stepping.dat” usingstp = read.table("stepping.dat", header=TRUE)dim(stp) # 30 6sapply(stp, class)# Order Block Height Frequency RestHR HR# "integer" "integer" "integer" "integer" "integer" "integer"stp$Block = factor(stp$Block)stp$Height = factor(stp$Height, labels=c("Low","High"))stp$Frequency = factor(stp$Frequency, labels=c("Low","Med","High"))summary(stp)(a) Run with(stp, table(Height, Frequency, Block)) and turn in a state-ment of what you observe.(b) Make a similar table without Block, and turn in a statement of what youobserved including either the term “balanced” or “unbalanced”.(c) Run the additive 3-way ANOVA model for the HR outcome with all threeexplanatory factors. Try different orderings of the variables paying specialattention to the SS values and F values (as an easier to read surrogate for thep-values). Explain the pattern when i) comparing Block+Frequency+Heightto Frequency+Block+Height and ii) comparing Block+Frequency+Height toBlock+Height+Frequency.(d) Taking any of the tables of part c, we can see the dfW= 21, SSW= 1169.2,and M SW= 55.7, where W stands for “within groups”. Some programs in-clude lines for Between Groups and Total. What would the values be fordfB, SSB, MSB, dfT, and SST?(e) Logically the interaction to be most concerned about is that the effect of achange in step height on heart rate depends on (changes with) the specificlevel of treatment. Fit that model (including blocks as the first factor) andturn in the p-value for the interaction, and your conclusion about interaction.(f) When we have a term in a model that is not statistically significant, it is helpfulto get a CI on the effect estimates to allow subject matter experts to determinewhether a practically significant effect is likely.We will examine whether the step height effect is statistically different for lowvs. high frequency. First run2mi=aov(HR~Block+Frequency+Height+Frequency:Height, stp)coefficients(mi)to see the estimated coefficients. Next think carefully about how you would usethe coefficients to calculate the expected HR for conditions FH&HH, FH&HL,FL&HH, and FL&HL, where, e.g., FH&HH means frequency is high and stepheight is high.By examining the interaction plot, you can see that the quantity of interest is(FH&HH-FH&HL) - (FL&HH-FL&HL), which simplifies nicely in this case tobFrequencyHigh:HeightHigh, with an estimate of 9.75. To make a CI, we need to getthe standard error of this quantity, using eithersqrt(vcov(mi)["FrequencyHigh:HeightHigh", "FrequencyHigh:HeightHigh"])or summary.lm(mi), which both give 6.163. The pertinent df is from the Resid-ual line: 19. The “plus or minus” comes from qt(0.975, 19).Give a careful statement about our confidence in the size of the interactionusing the confidence interval (rounded to whole numbers) in a form that couldbe understood by an exercise physiologist. Start with “We are 95% confidentthat ....”.(g) Turn in the R summary(aov()) table for the additive 3-way ANOVA modeland your conclusions about whether or not step height and frequency haveeffects on heart rate.(h) The client asks for the following planned contrast hypotheses: to test whetherthe “high frequency heart rate differs from the average of the medium and lowfrequencies” and whether the “medium frequency heart rate differs from thelow frequency.” Using the additive model, construct the coefficients and carryout the contrast tests using fit.contrasts() in package “gmodels”. Turn in theR code, the R output, and a brief summary of your conclusions, being sure tocorrectly explain the sign of any significant effect(s).(i) Explain why it does not make sense to construct contrasts for Height (hint:the answer is statistical, not from the subject matter.)3. Writing exercise #1 (15 points) Turn this in on a separate non-stapled pieceof paper with your name on it.You are the statistician for the Institute of Global Oceanographic Research (IGOR).Your boss, Dr. Frankenstein, is an intelligent, but very busy woman who had onlyone statistics course 20 years ago. You must write a one page (one side only)executive report for her, summarizing how you analyzed the algal regrowth dataand what you conclusions are.