1. (Ch. 19, # 7; 8 marks) In a study designed to examine the effects of adding oats to the typical American diet, individuals were randomly divided into two different groups. Twice a day, the first 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 affect this response include diet group, serum cholesterol level at the start of the study, body mass index, and gender. The estimated coefficients and standard errors from the multiple regression model containing these four explanatory variables are displayed below.

Variable Coefficient Standard Error Diet Group Diet Group -11.25 4.33 Baseline Cholesterol 0.85 0.07 Body Mass Index 0.23 0.65 Gender -3.02 4.42

(a) (4 marks, one for the test of each coefficient) Conduct tests of the null hypotheses that each of the four coefficients in the population regression equation is equal to 0. At the 0.05 level of significance, which of the explanatory variables have an effect on serum cholesterol level eight weeks after the start of the study?

(b) (1 mark) If an individuals body mass index were to increase by 1kg/m2while the values of all other explanatory variables remained constant, what would happen to his or her serum cholesterol level?

(c) (1 mark) If an individuals body mass index were to increase by 10 kg/m2while 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?

2. (Ch. 19, #8; 11 marks)For the population of low birth weight infants, a significant linear relationship was found to exist between systolic blood pressure and gestational age (dataset 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 five-minute apgar score for each infant. (The apgar score is an indicator of a childs general state of health five minutes after it is born; although it is actually an ordinal measurement, it isoften treated as if it were continuous.)

(a) (2 marks) Construct a two-way scatter plot of systolic blood pressure versus five-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 ageand apgar score as the explanatory variables, fit the least-squares model

y = a + 1x1 + 2x2.

Interpret 1, the estimated coefficient of gestational age. What doesit mean in words? Similarly, interpret 2, the estimated coefficient of fiveminuteapgar 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 five-minute apgar score is 7?

(d) Skip part (d)

(e) (2 marks) Test the null hypothesisH0 : 2 = 0at the 0.05 level of significance. What do you conclude?

(f) (2 marks) Comment on the magnitude of R2. Does the inclusion of fiveminuteapgar score in the model already containing gestational age improveyour ability to predict systolic blood pressure?

(g) (2 marks) Construct a plot of the residuals versus the fitted values of systolic blood pressure. What does this plot tell you about the fit of the model to the observed data?

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 and0 a female to the model that contains gestational age. Given two infantswith identical gestational ages, one male and the other female, which wouldtend 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 different effect on systolic blood pressure depending on the gender of the infant?

LOWBWT DATASET

 sbp sex tox grmhem gestage apgar5 43 1 0 0 29 7 51 1 0 0 31 8 42 0 0 0 33 0 39 0 0 0 31 8 48 0 1 0 30 7 31 1 0 1 25 0 31 1 1 0 27 7 40 0 0 0 29 9 57 0 0 0 28 6 64 0 1 0 29 9 46 0 0 0 26 7 47 0 0 1 30 6 63 0 0 0 29 8 56 0 0 0 29 1 49 1 0 0 29 8 87 1 0 0 29 7 46 0 0 0 29 8 66 0 0 0 33 8 42 0 1 0 33 8 52 0 0 0 29 7 51 1 0 0 28 7 47 0 0 0 30 9 54 1 0 0 27 4 64 1 0 0 33 9 37 0 0 0 32 7 36 1 1 0 28 3 45 0 0 1 29 7 39 1 0 0 28 7 29 0 0 0 29 4 61 0 0 0 30 3 53 1 0 0 31 7 64 0 0 0 30 7 35 0 0 0 31 6 34 1 0 0 29 9 62 0 0 0 27 7 59 0 0 0 27 8 36 1 0 0 27 9 47 0 0 0 32 8 45 1 0 1 31 2 62 0 0 1 28 5 75 1 1 0 30 7 44 1 0 0 29 0 39 1 0 0 28 8 48 0 0 1 31 7 43 0 1 0 27 6 19 0 0 1 25 4 63 1 0 0 30 7 42 1 0 0 28 6 44 0 0 0 28 9 25 0 0 0 25 8 26 0 0 0 23 8 27 1 0 0 27 9 35 1 0 0 28 8 40 1 0 0 27 7 44 0 0 0 27 6 66 1 0 0 26 8 59 0 0 0 25 3 24 0 0 0 23 7 40 0 0 1 26 3 49 0 0 0 24 5 53 1 1 0 29 9 45 0 0 0 29 9 50 1 0 1 27 8 64 1 0 0 30 7 48 0 0 0 30 6 48 0 0 1 32 4 58 0 1 0 33 7 67 0 0 0 27 8 40 0 0 1 31 8 48 0 0 0 26 8 36 1 0 0 27 5 44 1 0 0 27 6 53 0 1 0 35 9 45 0 1 0 28 6 54 1 0 0 30 8 44 1 1 0 31 2 42 1 0 0 30 5 50 0 0 0 27 0 48 0 0 0 25 5 29 0 0 1 25 5 30 0 0 1 26 2 36 0 0 0 29 0 44 0 0 0 29 0 46 0 1 0 34 9 51 1 1 0 30 4 51 1 0 0 29 5 43 1 1 0 33 7 48 1 0 0 30 5 52 1 0 0 29 8 43 1 0 0 24 6 42 1 1 0 33 8 48 1 0 1 25 5 49 0 1 0 32 8 62 1 1 0 31 7 45 1 0 0 31 9 51 0 1 1 31 6 52 1 0 0 29 8 47 1 1 0 32 5 40 0 1 0 33 8 50 0 0 0 28 7