Statistics 305 Homework 5 Due: Dec 02, 16:30 1. (Ch. 19, # 7; 8 marks) In a study designed to examine the eects of adding oats to the typical American diet, individuals were randomly divided into two dierent groups. Twice a day, the rst group substituted oats for other foods containing carbohydrates; the members of the second group did not make any changes to their diet. One outcome of interest is the serum cholesterol level of each individual eight weeks after the start of the study. Explanatory variables that might aect this response include diet group, serum cholesterol level at the start of the study, body mass index, and gender. The estimated coecients and standard errors from the multiple regression model containing these four explanatory variables are displayed below. Variable Diet Group Baseline Cholesterol Body Mass Index Gender Coecient Standard Error -11.25 4.33 0.85 0.07 0.23 0.65 -3.02 4.42 (a) (4 marks, one for the test of each coecient) Conduct tests of the null hypotheses that each of the four coecients in the population regression equation is equal to 0. At the 0.05 level of signicance, which of the explanatory variables have an eect on serum cholesterol level eight weeks after the start of the study? (b) (1 mark) If an individual's body mass index were to increase by 1kg/m2 while the values of all other explanatory variables remained constant, what would happen to his or her serum cholesterol level? (c) (1 mark) If an individual's body mass index were to increase by 10 kg/m2 while the values of all other explanatory variables remained constant, what would happen to his or her serum cholesterol level? (d) (2 marks) The indicator variable gender is coded so that 1 represents a male and 0 a female. Who is more likely to have a higher serum cholesterol level eight weeks after the start of the study, a man or a woman? How much higher would it be, on average? 1 2. (Ch. 19, #8; 11 marks) For the population of low birth weight infants, a signicant linear relationship was found to exist between systolic blood pressure and gestational age (data set lowbwt). The measurements of systolic blood pressure are saved under the variable name sbp, and the corresponding gestational ages under gestage. Also contained in the data set is apgar5, the ve-minute apgar score for each infant. (The apgar score is an indicator of a child's general state of health ve minutes after it is born; although it is actually an ordinal measurement, it is often treated as if it were continuous.) (a) (2 marks) Construct a two-way scatter plot of systolic blood pressure versus ve-minute apgar score. Does there appear to be a linear relationship between these two variables? (b) (2 marks) Using systolic blood pressure as the response and gestational age and apgar score as the explanatory variables, t the least-squares model y = a + 1 x1 + 2 x2 . Interpret 1 , the estimated coecient of gestational age. What What does it mean in words? Similarly, interpret 2 , the estimated coecient of veminute apgar score. (c) (1 mark) What is the estimated mean systolic blood pressure for the population of low birth weight infants whose gestational age is 31 weeks and whose ve-minute apgar score is 7? (d) Skip part (d) (e) (2 marks) Test the null hypothesis H0 : 2 = 0 at the 0.05 level of signicance. What do you conclude? (f) (2 marks) Comment on the magnitude of R2 . Does the inclusion of veminute apgar score in the model already containing gestational age improve your ability to predict systolic blood pressure? (g) (2 marks) Construct a plot of the residuals versus the tted values of systolic blood pressure. What does this plot tell you about the t of the model to the observed data? 2 3. (Ch. 19, #9; 4 marks) The data set lowbwt also contains sex, a dichotomous random variable designating the gender of each infant. (a) (2 marks) Add the indicator variable sex - where 1 represents a male and 0 a female -to the model that contains gestational age. Given two infants with identical gestational ages, one male and the other female, which would tend to have the higher systolic blood pressure? By how much, on average? (b) (2 marks) Add to the model a third explanatory variable that is the interaction between gestational age and sex. Does gestational age have a dierent eect on systolic blood pressure depending on the gender of the infant? 3