1.The American Heart Association collects data on the risk of strokes. A 10-year study provided data on how age (X1), blood pressure (X2), and smoking (X3) relate to the risk of strokes (Y). Assume that the following data are from a portion of this study. Risk is interpreted as the probability (times 100) that the patient will have a stroke over the next 10-year period. For the smoking variable (X3), define a dummy variable with 1 indicating a smoker and 0 indicating a nonsmoker.
(a) Construct a scattergram between Risk (Y) and the Age (X1). Does there appear to be a linear relationship?
(b) Estimate a simple linear regression between Y and X1. Briefly evaluate the estimated simple regression equation.
(c) Construct a scattergram between Risk (Y) and the Blood Pressure (X2). Does there appear to be a linear relationship?
(d) Estimate a simple linear regression between Y and X2. Briefly evaluate the estimated simple regression equation.
(e) Estimate a multiple linear regression between Y and X1, X2 & X3. Briefly evaluate the estimated multiple regression equation.
(f) Provide an interpretation of each estimated regression coefficient in part e.
(g) Is smoking a significant factor in the risk of a stroke based on the results of the estimated regression equation in part e? Briefly explain.
(h) Which estimated regression equation provides the "Best Fit" to the data? Briefly explain.
(i) Utilize the equation of "Best Fit" (see part h) to predict the Risk of a Stroke over the next 10 years for a 68-year old smoker who has blood pressure of 175? What action might the physician recommend for this patient?
2. Continuation of previous problem. The takeaway from the answers is that the multiple regression with square footage (x1),number of bedrooms (x2), and number of bathrooms (x3), and distance from City Center (x4) is a pretty good regression equation with an Rsquare =.384 Adjusted Rsquare = .359, the individual variables are all statistically significant and the F-Test indicates global statistical significance.
Rsquare of only .384 indicates that the estimated multiple regression equation is not explaining about 61.6 percent of the variation of sales price (dependent variable). The search continues for additional statistically significant explanatory variables.
If the home has a pool may be relevant in explaining the variation in the sales price of homes. A potential fifth explanatory variable has been provided (x5) to put together a multiple regression equation. Since presence of a pool (Yes or No) is a qualitative variable, a dummy variable has been set where a home with a pool (Yes) = 1 and pool (no) = 0.
(a) What is your expectation about the direction of the relationship between sales price and the pool dummy variable based on the coding scheme (Pool = 1, No Pool = 0)?
(b) Estimate the multiple regression equation (include square footage, number of bedrooms, number of bathrooms, distance from the center of the city, pool dummy variable).
(c) Conduct individual hypothesis tests (alpha=.05) to determine if any of the explanatory variables can be dropped from the estimated multiple regression equation.
(d) Conduct a global hypothesis test (F-Test) to verify the overall statistical significance (p-value =.05) of the estimated multiple regression equation in part b.
(e) Provide an interpretation of the estimated coefficient on the pool dummy variable.
(f) If any variables should be dropped from the multiple regression equation, re-run and repeat parts c and d and e (if applicable).
(g) Provide an overall written evaluation of your final multiple regression equation justifying your decision. Is the explanatory dummy variable pool worthwhile addition to the multiple regression equation?
(h) Based on the final regression equation, forecast the sales price for a home which is 3,950 square feet has 4 bedrooms and 3 bathrooms, is located 21 miles from the center of the city and has a pool (dummy variable 1).
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