7: Correlation & Linear Regression and Multiple Linear Regression

Question 1

___ is a group of techniques to measure the relationship between two variables.

Answer 1

Correlation analysis

Question 2

The typical first step of a correlation analysis is a ___.

Answer 2

Scatter diagram

Question 3

___ is a measure of strength of association between two interval level or ratio level variables.

Answer 3

Correlation coefficient

Question 4

A correlation coefficient can range from ___ to ___.

Answer 4

-1.00 to 1.00

Question 5

If there is absolutely no relationship between the two sets of variables, Pearson’s r is ___.

Answer 5

Zero

Question 6

___ is an equation that expresses the linear relationship between two variables.

Answer 6

Regression equation

Question 7

___ is a mathematical procedure that uses the data to position a line with the objective of minimizing the sum of the squares of the vertical distances between the actual y values and the predicted values of y.

Answer 7

Least squares principle

Question 8

___ is a mathematical procedure that uses the data to position a line with the objective of minimizing the sum of the squares of the vertical distances between the actual y values and the predicted values of y.

Answer 8

Standard error of estimate (Sy⋅x)

Question 9

___ is the proportion of the total variation in the dependent variable Y that is explained, or accounted for, by the variation in the independent variable X.

Answer 9

Coefficient of determination (r²)

Question 10

Regression analysis provides two statistics to evaluate the predictive ability of a regression equation: ___ and ___.

Answer 10

1. The standard error of the estimate (Sy⋅x)

2. The coefficient of determination (r²)

Question 11

What are the 4 Assumptions Underlying Linear Regression?

1. ___
2. ___
3. ___
4. ___

Answer 11

1. For each value of x, there are corresponding y values. These y values follow the normal distribution.
2. The means of these normal distributions lie on the regression line.
3. The standard deviations of these normal distributions are all the same. The best estimate we have of this common standard deviation is the standard error of estimate (sy·x).
4. The y values are statistically independent. This means that in selecting a sample, a particular x does not depend on any other value of x. This assumption is particularly important when data are collected over a period of time. In such situations, the errors for a particular time period are often correlated with those of other time periods.

Question 12

A relationship exists between the predicted values yˆ and the standard error of estimate (sy⋅x).

___ will include the middle 68% of the observations.
___ will include the middle 95% of the observations.
___ will include virtually all the observations.

Answer 12

yˆ±sy⋅x
yˆ±2sy⋅x
yˆ±3sy⋅x

Question 13

In constructing intervals:

The first interval estimate is called a ___ interval.
The second interval estimate is called a ___ interval.

Answer 13

Confidence interval

Prediction interval

Question 14

The ___ variable is scaled on the Y-axis and is the variable being estimated.

Answer 14

Dependent

Question 15

The ___ variable is scaled on the X-axis and is the variable used as the predictor.

Answer 15

Independent

Question 16

The number of degrees of freedom associated with the error term is equal to ___

Answer 16

The total degrees of freedom, n − 1

Question 17

___ is the percent of variation in the dependent variable, y, explained by the set of independent variables, x1, x2, x3, …

Answer 17

Coefficient of multiple determination

Question 18

The characteristics of the coefficient of multiple determination are:

1. ___
2. ___
3. ___
4. ___

Answer 18

1. It is symbolized by a capital R squared. In other words, it is written as R2 because it is calculated as the square of a correlation coefficient.
2. It can range from 0 to 1. A value near 0 indicates little association between the set of independent variables and the dependent variable. A value near 1 means a strong association.
3. It cannot assume negative values. Any number that is squared or raised to the second power cannot be negative.
4. It is easy to interpret. Because R2 is a value between 0 and 1, it is easy to interpret, compare, and understand.

Question 19

___ is a test used to determine if any of the set of independent variables has regression coefficients different from zero.

Answer 19

Global test

Question 20

To test the null hypothesis that the multiple regression coefficients are all zero, we employ ___.

Answer 20

The F distribution

Question 22

The decision rule can be based on either of two methods:

1. ___
2. ___

Answer 22

(1) comparing the test statistic to a critical value

(2) calculating a p-value based on the test statistic and comparing the p-value to the significance level

Question 23

The critical value method using the F-statistic requires three pieces of information:

1. ___
2. ___
3. ___

Answer 23

(1) the numerator degrees of freedom
(2) the denominator degrees of freedom
(3) the significance level

Question 24

The ___ distributions of the coefficients follow the t distribution with n − (k + 1) degrees of freedom.

Answer 24

Sampling

Question 25

The sampling distributions of the coefficients follow the ___ degrees of freedom.

Answer 25

n − (k + 1)

Question 26

The linearity assumptions are:

1. ___
2. ___
3. ___

Answer 26

1. Residuals are plotted on the vertical (y) axis and centered at zero.
2. Residual plots show a random distribution of positive and negative values across the horizontal (x) axis.
3. Plots are scattered and there is no obvious pattern.

Question 27

___ compares all possible models using a specified set of predictors, and displays the best-fitting models that contain one predictor, two predictors, and so on.

Answer 27

Best subset regression

Question 28

A graph that helps to evaluate the assumption of normally distributed residuals is called a ___ and is often included in statistical software. It supports the assumption of normally distributed residuals.

Answer 28

Normal Probability Plot

Question 29

___ is the variation around the regression equation is the same for all of the values of the independent variables.

Answer 29

Homoscedasticity

Question 31

___ is a phenomenon in which one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy.

Answer 31

Multicollinearity

Question 32

Several clues that indicate problems with multicollinearity include the following:

1. ___
2. ___
3. ___

Answer 32

1. An independent variable known to be an important predictor ends up having a regression coefficient that is not significant.
2. A regression coefficient that should have a positive sign turns out to be negative, or vice versa.
3. When an independent variable is added or removed, there is a drastic change in the values of the remaining regression coefficients.

Question 33

A general rule is… if the correlation between two independent variables is___, then there likely ___ a problem using both of the independent variables.

Answer 33

between −0.70 and 0.70

is not

Question 34

___ is the term used for when successive residuals are correlated

Answer 34

Autocorrelation

Question 35

___ is a step-by-step method to determine a regression equation that begins with a single independent variable and adds or deletes independent variables one by one. Only independent variables with nonzero regression coefficients are included in the regression equation.

Answer 35

Stepwise regression

Question 35

___ is a variable in which there are only two possible outcomes. For analysis, one of the outcomes is coded a 1 and the other a 0.

Answer 35

Dummy variable

Question 36

The null hypothesis and the alternate hypothesis regarding the Significance of the Correlation Coefficient are:

H0: ___
H1: ___

Answer 36

H0: ρ = 0 (The correlation in the population is zero.)
H1: ρ ≠ 0 (The correlation in the population is different from zero.)

Question 36

The number of degrees of freedom in the regression is equal to ___ in the multiple regression equation.

Answer 36

The number of independent variables

Question 36

The regression degrees of freedom is ___.

Answer 36

k

Question 37

The residual or error degrees of freedom is ___, and is the same as n − (k + 1).

Answer 37

(n − 1) − k

Question 39

The total variation of the dependent variable, y, is divided into two components:

1. ___
2. ___

Answer 39

(1) regression, or the variation of y explained by all the independent variables
(2) the error or residual, or unexplained variation of y