For the theory section, show your work and do not round off your middle work. 

Keep your final answer to four (4) decimal places, where relevant.

1. In a regression model involving 37 observations, the following estimated regression equation was obtained:

         y = 34 + 24x1 + 36x2 + 43x3

It is also known that SS Total = 500 and SS Residual = 175.

a. What is the coefficient of determination?

b. Compute the adjusted coefficient of determination.

c. What is the MS residual?

d. What is the computed F statistic? 

e. Comment on how good the fit of the model is.

2. The table shows part of a computer output based on 18 observations. 

        Coefficient Standard error

Constant 36.325 2.380

x1                 -5.893 2.150

x2                    13.297 8.768

a. What is the estimated regression line?

b. Interpret each of the three coefficients in the estimated regression equation.

c. Estimate y when x1 = 14 and x2 = 39. 

d. Test the significance of β1 at the 0.05 level of significance. 

e. Should x1 be dropped from the model? Justify your answer.

f. Test the significance of β2 at the 0.05 level of significance.

g. Should x2 be dropped from the model? Justify your answer.

3. In multiple regression

a. what is multicollinearity? What is its effect on fitting multiple regression models?

b. what does the variance inflation factor measure?

c. what are dummy variables?

d. what is the adjusted R2?

4. Consider a regression study involving one dependent variable y, one quantitative independent variable and one qualitative independent variable with three possible levels.

a. Write a multiple regression equation involving the variables.

b. Interpret the coefficient (β values) in your equation from part a by solving this equation for y when:

i) the quantitative independent variable equals 10 and the qualitative independent variable equals level 1

ii) the quantitative independent variable equals 10 and the qualitative independent variable equals level 2

iii) the quantitative independent variable equals 10 and the qualitative independent variable equals level 3

5. Suppose you are fitting y as a function of x1, x2, x3, x4, and x5. Describe the backward elimination method of variable selection.

6. How is Mallows’ Cp statistic calculated and how is it used in selecting a subset of predictor variables? 

7. a.  Under what circumstances would logistic regression be used?   Why would use of linear regression not be ideal under such circumstances?

b. It is known that some of the risk factors that affect heart disease are age, cholesterol level and weight. To estimate the probability of heart attack among smokers, a logistic regression model was developed by a researcher:

ln(y) = β0 + β1x1 + β2x2 + β3x3 + β4x4,

where 

y is the odds ratio of suffering attack

x1 is the age

x2 is the number of pounds of overweight

x3 is the cholesterol level

x4 is the number of cigarettes smoked.

The coefficients are

β0 = -3.12

β1.= 0.01028

β2 = 0.00673

β3 = 0.00316 

β4 = 0.00942

i. Why is the logistic regression model appropriate in this case?

ii. What is the probability of heart attack for the individuals shown in the table below?

Individual    Age Pounds      Cholesterol Cigarettes 

                             overweight        level                  per day

A                   55          3                170                   4

B                   45         15                 210                   0

C                   36    45                 190             15

D              69         30                 240                 20

8. Provide brief explanations for

a. Cook’s distance measure.

b. autocorrelation (also known as serial correlation).

c. the Durban-Watson test statistic.