Question

A university would like to develop a regression model to predict the point differential for games played by its men's basketball team. A point differential is the difference between the final points scored by two competing teams. A positive differential is a win for the university's team and a negative differential is a loss. For a random sample of games, the point differential (y) was calculated, along with the number of assists (x1), rebounds (x2), turnovers (x3) and personal fouls (x4).Complete parts a through f below.

Game   Point_Differential   Assists   Rebounds   Turnovers   Personal_Fouls
1   16   14   38   12   10
2   45   22   46   11   11
3   11   19   45   11   7
4   19   9   34   17   17
5   16   17   36   13   20
6   14   20   36   18   12
7   11   15   26   10   10
8   5   13   28   12   18
9   -3   6   33   12   23
10   -5   14   30   12   15
11   -10   4   22   8   21
12   23   16   39   8   22
13   6   12   38   13   17
14   19   14   39   13   26
15   -10   12   24   9   19
16   38   19   44   10   19
17   2   13   31   7   17
18   8   10   34   21   33
19   10   10   34   20   20
20   -13   11   26   17   38
21   -9   9   26   9   30
22   -2   16   32   15   24

a) Construct a regression model using all four independent

variables.

y= _ + (_)x1 + (_)x2 + (_)x3 + (_)x4

b) Test the significance of each independent variable using a=.05

Identify the general form of the null and alternative hypotheses, where βj is the population regression coefficient for the jth independent variable.

H₀: B_{j =}

H₁:Bj=

c) Interpret the p-value for the coefficient of each independent variable. Find the p-value for the coefficient of each independent variable.

d) Construct a 95​% confidence interval for the regression coefficients for the Personal Fouls independent variable and interpret the meaning. A 95% confidence interval for the coefficient of the Personal Fouls independent variable is ___ to ___ The Data

Answer 1

(a) Multiple Regression

The proposed multiple linear regression model :

Where Y: the point differential, X₁ : no. of assists; X₂ : number of rebounds; X₃ : number of turnovers; X₄ : number of personal fouls; and : the residual term. Therefore, here the response variable is the risk of having a stroke while the other three variables described above are the predictor variables. Also, = intercept term and = slope.

Assuming, that the sample was randomly drawn, the random variables used in the equation are independent of each other, and the residuals follow an i.i.d. N(0,1). Following the ordinary least square (OLS) method, the estimates are determined as below:

where the slope and intercepts are found through the OLS method based on the sample of x₁,x₂, x₃ and x₄.

The regression equation is determined to be as follows:

point differential = -46.9 + 0.56*number of assists + 1.53*number of rebounds - 0.06*number of turnovers - 0.15*number of personal fouls

(b) Test of Significance

Consider the general hypothesis test for the coefficients of each independent variable in the equation as

(c) Interpretation of the p-values

For number of assists the p-value of the respective test is 0.38 > 0.05

For number of rebounds the p-value of the respective test is 0.0013 < 0.05

For number of turnovers the p-value of the respective test is 0.91 > 0.05

For number of personal fouls the p-value of the respective test is 0.65 > 0.05

Therefore, only in one case the null hypothesis can be rejected at 5% significance level.

In other words, the tests of significance show that only number of rebounds is statistically significant to predict point-differential.

(d) 95% Confidence Interval

The 95% confidence interval for number of fouls is given as (-0.803,0.5107) and this shows that the predictor, number of fouls, is not statistically significant to predict the point differential.