chapter 14: multiple regression and model building

Question 1

what are multiple regression models?

Answer 1

regression models that employ more than one independent variable

Question 2

why is it possible for the multiple regression model formula to be like the:

y = B0 + B1x1 + B2x2 + E

why are there 2 xs?

Answer 2

because the mean level (Uy) now is B0 + B1x1 + B2x2

this basically means that there are two different independent variables that can correlate or “Influence” the dependent variable “y”

E still remains the error term that causes y to deviate from the mean level

Question 3

what is the new name for the mean level:

Uy = B0 + B1x1 + B2x2

Answer 3

the plane of means

its in a three dimensional space

Question 4

what is B0 in Uy = B0 + B1x1 + B2x2?

Answer 4

it is still the y intercept

Question 5

what is B1 in Uy = B0 + B1x1 + B2x2?

Answer 5

the regression parameter for the variable x1

the slope of the plane of the x1 direction

Question 6

what is B2 in Uy = B0 + B1x1 + B2x2?

Answer 6

the regression parameter for the variable x2

the slope of the plane of the x2 direction

Question 7

what is the error term E in Uy = B0 + B1x1 + B2x2?

Answer 7

the error term

what describes the effects on y other than x1 and x2

Question 8

what is the formula of the point estimate or prediction of

y = B0 + B1x1 + B2x2 + E

what is the name of such equation

Answer 8

y^ = b0 + b1x1 + b2x2

called the least squared plane, the estimate of the plane of means

Question 9

what is there no error term when we use

y^ = b0 + b1x1 + b2x2

to predict a point of

y = B0 + B1x1 + B2x2 + E

Answer 9

the error term has a 50% chance of being positive and 50% chance of being negative

Question 10

what is the residual?

Answer 10

the difference between the observes and predicted values

Question 11

what is SSE

Answer 11

the unexplained variation

the sum of the squared residuals

Question 12

what is the multiple coefficient of determination?

Answer 12

the proportion of the Total variation in the n observed values of the dependent variable that is explained by the overall regression model

R^2

R^2 = explained variation / total variation

Question 13

what is the multiple correlation coefficient

Answer 13

R

Question 14

what is the adjusted R^2

Answer 14

the adjusted multiple coefficient of determination used to avoid overestimating the importance of independent variables

adjusted R^2 =

(R^2 - (k / (n - 1))) * ((n - 1) / (n - (k + 1)))

n is the number of observations

k the number of independent variables in the model

Question 15

what are the four assumptions of the error term values in the multiple regression model?

Answer 15

1. at any given combination of x1, x2, …, xk, the population of potential error terms has a mean value of 0
2. constant variance assumption
3. normality assumption
4. independence assumption

Question 16

what is the error term constant variance assumption?

Answer 16

population of error term values has a variance that does not depend on the combination of values of x1, x2, …, xk

the different population of potential error terms corresponding to different combinations of values x1, x2, …, xk have equal variances

the constant variance is the population variance

Question 17

what is the error term constant normality assumption?

Answer 17

at any given combination of x1, x2, …, xk, the population of potential error terms has a normal distribution

Question 18

what is the error term constant independence assumption?

Answer 18

any one value of the error term E is statistically independent of any other value of E

an error term of a certain y has nothing to do with an error term of another y

Question 19

what is the point estimate of the constant variance of the different populations of error terms?

formula too

Answer 19

the mean square error

s^2

s^2 = SSE / (n - (k + 1))

Question 20

what is the point estimate of the standard deviation of the different populations of error terms?

formula too

Answer 20

the standard error

s

s = (SSE / (n - (k + 1)))^(1/2)

Question 21

in the mean square error and the standard error (the point estimate of the constant variance and the standard deviation of the different populations of error terms),

what is the meaning of the following

n - (k + 1)

Answer 21

degrees of freedom associated with SSE

Question 22

is testing the significance of the relationship between y and x1, x2, …, xk a proper way of assessing the utility of the regression model?

Answer 22

yeeee

Question 23

how do you test the significance of the relationship between y and x1, x2

Answer 23

with the F test

Question 24

what is the null hypothesis (H0) of the the significance of the relationship between y and x1, x2, …, xk

Answer 24

H0: B1 = B2 = … = Bk = 0

none of the independent variables x1, x2, … xk are significantly related to y

the regression relationship is not significant

Question 25

what is the alternative hypothesis (H0) of the the significance of the relationship between y and x1, x2, …, xk

Answer 25

Ha: at least one of B1, B2 … Bk =/= 0

at least one of the independent variables x1, x2, … xk is significantly related to y

the regression relationship is significant

Question 26

how do you calculate the F of the F statistic

Answer 26

F =

(explained variation) / k
____________________________
((unexplained variation) / (n - (k + 1)))

Question 27

how do the R^2 and adjusted R^2 differ?

Answer 27

R̅^2 differs from R^2 by taking into consideration the number of independent variables in the model

Using R̅^2 helps avoid overestimating the importance of the independent variables

Question 28

why would we test the significance of a single independent variable?

Answer 28

to gain further information of which independent variables significantly affect y?

Question 29

when you test the significance of a single independent variable, how do you refer to it?

what else must you assume?

Answer 29

xj

you have to assume it is multiplied by the parameter Bj

Question 30

what is the null hypothesis when you test xj?

Answer 30

Bj = 0

here, we say that xj is not significantly related to y

Question 31

what is the alternate hypothesis when you test xj?

Answer 31

Bj =/= 0

here, we say that xj is significantly related to y in the regression model under consideration

Question 32

what is the sbj

Answer 32

the standard error of the estimate bj

the point estimate of the population standard deviation of bj

Question 33

what test do you use to test the significance of xj?

Answer 33

the t test

Question 34

what is the formula of the t test to to test the significance of xj?

Answer 34

t = bj / sbj

Question 35

using the t test to to test the significance of xj, when do we reject Ho in favor of Ha?

Answer 35

t > t alpha

p value < significance value