5. Remember the form of the prediction interval and the form of the standard error of the prediction. For each part, state what will happen to the prediction interval (stay the same, be wider, or be narrower). a. The sample size is increased. b. If X}, gets closer to X, the average of the X covariate. c. If the variability of the response variable decreases. d. The average of the response is increased. 6. The sum of squares for the regression model, SSR, for the regression of Y on X was 110, and the sum of squared errors, SSE, was 40. Calculate and interpret the value of Hz. 7. We will use the Pulse.txt dataset for this question. This dataset is from an observational study where subjects had their resting pulse rate (beats per minute), height (in inches), weight g. Using the model output from part e., test the coefcient on Smoking yes (X2) being equal to 0 or not. Write the null and alternative hypothesis, test statistic, p-value, and make a conclusion in context of the study (be careful of your wording). h. Can we conclude that smoking causes lower resting pulse rates? Why or why not? 1. Create a scatterplot where X=Weight and Y=Re5ting pulse rate, and have different colors and symbols for the different smoking groups. Interpret this scatterplot. j. Now add an interaction term between smoker yes and weight to your model from part c., and write out the population model. k. Fit the model from part j. Write out the estimated regression equation. 1. What is the R2 value and 65 of this interaction model, and how do these values compare to the model that was t in part e. ? (in pounds), smoking status (1 for smoker and 0 for non-smoker), and active pulse rate (beats per minute) recorded. We will use resting pulse rate as the response variable Y, and explanatory variables of weight as X1 and smoking status as X2. Run the following line in R after you have imported the data set to create a new variable called "Smoker" with categories Yes and No. pulse$Smoker = ifelse(pulse$Smoke==1, "Yes", "No") a. Fit a model in R using response variable Y and only the covariate X2 of smoking status. Test if the coefficient on X2 is equal to 0. State the null and alternative hypothesis, the test statistic, p-value and make a conclusion. b. Now conduct a two sample t-test with equal varaince in R, where the response is resting pulse rate and the two groups/samples are smokers and non-smokers. How does this test correspond to the test from part a. ? c. Write out the population model where we have response Y and covariates/explanatory variables X1 and X2 with no interaction term in context of the study. d. With respect to the effect of weight on resting pulse rate, what does it mean in context of the study that we do not have an interaction term (with smoking status) in the model from part c. ? Explain in a sentence or two. e. Fit the model from part c. in R and report the coefficient of determination (multiple R2 value) and report the estimate of O. f. How many observations are there in the dataset?m. Test whether the effect of weight on resting pulse rate differs for smokers and non-smokers. Write out the null and alternative hypothesis, the test statistic, pvalue, and make a conclusion in context of the problem. n. Go back to your model that was tted in part e. Is this a good tting model (based on the Rsquared value)? 0. Explain how Exercise amount could be a potential confounder in this study. p. Hom your model in part j., how would you test if weight should be in the model? Write only the null and alternative hypothesis