A 10-year study conducted by the AHA provided data on how age, blood pressure, a
ID: 3218109 • Letter: A
Question
A 10-year study conducted by the AHA provided data on how age, blood pressure, and smoking relate to the risk of strokes. Assume that the following data are from a portion of this study. Risk is interpreted as the probablility (x 100) that the patient will have a stroke over the next 10 year period. For the smoking variable, define a dummy variable with 1 indicating a smoker and 0 indicating a nonsmoker.
a. Develop an estimated regression equation that relates risk of a stroke to the person's age, blood pressure, and whether the person is a smoker.
b. Is the model significant (or worth keeping)? What hypothesis test shows whether the model is significant. What is the deciding factor to keep or discard the model?
c. How much of the variablity in risk of stroke can be explained by the model? What is this known as?
d. Is smoking a significant factor in the risk of a stroke? How do you know? Use alpha = 0.05
e. What is the probability of a stroke over the next 10 years for Art Speen, a 68 year old smoker who has blood pressure of 175? What action might the physician recommend for the patient?
risk Age Pressure Smoker 12 57 152 no 24 67 163 no 13 58 155 no 56 86 177 yes 28 59 196 no 51 76 189 yes 18 56 155 yes 31 78 120 no 37 80 135 yes 15 78 98 no 22 71 152 no 36 70 173 yes 15 67 135 yes 48 77 209 yes 15 60 199 no 36 82 119 yes 8 66 166 no 34 80 125 yes 3 62 117 no 37 59 207 yesExplanation / Answer
a. Develop an estimated regression equation that relates risk of a stroke to the person's age, blood pressure, and whether the person is a smoker.
Solution:
The estimated regression equation for the dependent variable risk based on the independent variables age, pressure and dummy variable smoker is given as below:
Risk = -91.76 + 1.08*Age + 0.25*Pressure + 8.74*Smoker
b. Is the model significant (or worth keeping)? What hypothesis test shows whether the model is significant? What is the deciding factor to keep or discard the model?
Solution:
For checking whether the model is statistically significant or not, we have to use the ANOVA F test. The deciding factor to keep or discard the model is the p-value associated with given F test statistic. For the given ANOVA table for the regression model, we are given a p-value as 0.0000002 which is very less and so we reject the null hypothesis that the given regression model is not statistically significant. This means we conclude that there is sufficient evidence that the given regression model is statistically significant.
c. How much of the variability in risk of stroke can be explained by the model? What is this known as?
Solution:
The variability in the dependent variable due to the independent variables in the regression analysis is known as the coefficient of determination or the R square. It is measured as the square of the multiple correlation coefficient. For the given regression model, the value of R square or the coefficient of determination is given as 0.8735, which means about 87.35% of the variation in the dependent variable risk is explained by the independent variables age, pressure and smoker.
d. Is smoking a significant factor in the risk of a stroke? How do you know? Use alpha = 0.05
Solution:
The smoking is a significant factor in the risk of a stroke because for the coefficient of the variable smoker, the p-value is given as 0.01 which is less than alpha value 0.05, so we reject the null hypothesis that the coefficient for the variable smoker is not statistically significant. This means we conclude that there is sufficient evidence that the coefficient of the variable smoker is statistically significant.
e. What is the probability of a stroke over the next 10 years for Art Speen, a 68 year old smoker who has blood pressure of 175? What action might the physician recommend for the patient?
Solution:
Here, we have to estimate the value of the risk for the age = 68, pressure = 175 and smoker = 1.
Risk = -91.76 + 1.08*Age + 0.25*Pressure + 8.74*Smoker
Risk = -91.76 + 1.08*68 + 0.25*175 + 8.74*1
Risk = 34.17
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