3. This question will use the pulse.txt dataset we have been using. Let the response Y be Rest(resting heart rate), and covariates X1 be Height in inches (Hgt), X2 be Weight in pounds (Wgt)and X3 smoking status (Smoke, 1 for smokers and 0 for non-smokers).a. Say we want a multiple linear regression model with all the covariates listed in it, along withan interaction between height and weight. Write out this population model.b. Why does it make sense to have an interaction between weight and height in the model?Explain.c. Fit the model from part a. in R and write out the estimated model. What is the adjustedR-squared value?d. What is the estimated SSE (sum of squared errors) of your model? Use the R output from thesummary(model), along with some formulas, to compute this.e. Test if your model has any significance. Write out the null and alternative hypothesis, the teststatistic (and what distribution it follows), p-value, and make a conclusion.f. Test the interaction term between height and weight. State the null and alternative hypothesisand p-value. What can you conclude with respect to the effect of height and weight on the responseof resting heart rate ?g. Now a researchers states that you do not need weight in the model in any way or form. Writeout the null and alternative hypothesis for this (write it out in terms of slope coefficients).h. Conduct the test from part g. Make a conclusion in context of the study.i. Show the output from R for the sequential sum of regressions table (keep the order of X1, X2, X3, andX1X2in your model). Does adding weight when height is already in the model add to the explanatory strength of the model?
j. Using the sequential sum of regression table from part i, what is the SSTO (sum of squared total) for this model?
k. Now say you have a model with only height in it. From your table in part i, what is the SSE, SSR, and SSTO of this model with only height as a covariate? Note: Do not run a new model, use the table from part i.
l. Explain in a few sentences how daily exercise amount could be a potential confounder in the model from part a.
m. Using the model from part a., create and present the scatterplot of the residual versus fittedvalues (fitted values go on the X-axis). Is there any evidence that our assumption of constantvariance (a single σ2 for the entire model) is invalid?n. Using the residual vs. fitted plot from part m., comment on the linearity assumption of themodel that was fit.o. Using the model from part a., now create a QQ plot of the residuals. Do we seem to have anyissues with our normality assumption for the errors?p. Using the model from part a., do a summary(data) to see the summary statistics of the weight(Wgt) explanatory variable. Does it make sense to use your model from part a. to predict the resting heart rate for someone who weight 350 pounds? Why or why not