Chapter 4 Describing the Relationship Between Two Variables

Question 1

Least Squares Regression

Answer 1

The procedure used to determine if there is a linear relationship or correlation between two variables.

Question 2

Least Squares Regression Line

Answer 2

The straight line that “best fits the data points” plotted in a scatter diagram. The line that has the least sum of its squared errors.

Question 3

Least Squares Regression Procedure

Answer 3

Step 1: Construct a scatter diagram (explanatory variable on horizontal axis, response variable on the vertical axis)
Step 2: Determine the mathematical measures of linearity and the equation of the Least Squares Regression Line
- Correlation Coefficient
- Coefficient of Determination
Step 3: Plot and examine the residuals (difference between the regression line and the actual data)

Question 4

Positively Associated (or Correlated)

Answer 4

Occurs whenever the value of one variable increases, and the value of the other variable increases also.
—- In other words, if the trend has a positive SLOPE, the variables are positively associated.

Question 5

Negatively Associated (or Correlated)

Answer 5

Occurs if whenever the value of one variable increases, the value of the other decreases.
—- In other words, if the trend has a negative SLOPE, the variables are negatively associated (or correlated).

Question 6

NOT Linearly Associated (or Correlated).

Answer 6

Occurs If the trend of the data points shows neither a positive or negative slope, but rather a more or less random pattern

Question 7

Linear Correlation Coefficient (LCC)

Answer 7

A measure of the strength and direction of the linear relation between two quantitative variables. Denoted as “r” for samples

Question 8

PROPERTIES of the LCC

Answer 8

-1 ≤ r ≤ 1
If r = +1, a perfect positive linear relation exists. (The closer to +1, the more positive the association)
If r = -1, a perfect negative linear relation exists. (The closer to -1, the more negative the association)
If r = 0, no evidence of linear relation.

Question 9

Coefficient of Determination

Answer 9

Measures the proportion of total variation in the response variable that is explained by the least-squares regression line. Denoted by R^2.
R^2 = (r) times (r)

Question 10

Residual

Answer 10

On a scatter plot, the difference between the observed value of y and the value of y on a candidate least squares regression line (y^). Denoted (y - y^).

Question 11

The Slope-Intercept Form of the Equation of a Straight Line Applied to Linear Regression

Answer 11

y^ = mx + b, where “x” is the explanatory variable, y^ the estimation (or prediction) of the response variable, “m” is the slope of the line and “b” is the y-intercept of the line.

Question 12

Properties of the Coefficient of Determination (R^2)

Answer 12

0 ≤ R^2 ≤ 1
If R2 = 1, the regression line explains 100% of the variation in the response variable.
If R2 = 0, the regression line has no value.

Question 13

Is the Linear Correlation Coefficient a resistant measure of linear association?

Answer 13

The LCC is not a resistant measure of linear association.

Question 14

Does an LCC near zero mean that there is no relation between two variables?

Answer 14

It just means that there is no linear correlation between the variables.

Question 15

Properties of the Linear Correlation Coefficient

Answer 15

The linear correlation coefficient is always between −1 and 1, inclusive. That is, −1≤r≤1.

If r=+1, then a perfect positive linear relation exists between the two variables. See Figure 4(a).

If r=−1, then a perfect negative linear relation exists between the two variables. See Figure 4(d).

The closer r is to +1, the stronger is the evidence of positive association between the two variables. See Figures 4(b) and 4(c).

The closer r is to −1, the stronger is the evidence of negative association between the two variables. See Figures 4(e) and 4(f).

If r is close to 0, then little or no evidence exists of a linear relation between the two variables. So a value of r close to 0 does not imply no relation, just no linear relation. See Figures 4(g) and 4(h).

The linear correlation coefficient is a unitless measure of association. So the unit of measure for x and y plays no role in the interpretation of r.

The correlation coefficient is not resistant. Therefore, an observation that does not follow the overall pattern of the data could affect the value of the linear correlation coefficient.

Question 16

Using the LCC to Determine Whether a Linear Relation Exists Between Two Variables

Answer 16

If the correlation coefficient is positive and greater than the critical value, then the variables are positively associated.

If the correlation coefficient is negative and less than the opposite of the critical value, then the variables are negatively associated.

Question 17

Time-Series Data

Answer 17

Data collected over a period of time (say from 2000 through 2017).

Question 18

Lurking Variable

Answer 18

An explanatory variable that was not considered in the study, but affects the response variable.

In addition, lurking variables are typically related to explanatory variables considered in the study.

Question 19

Least-Squares Regression Criterion

Answer 19

The least-squares regression line minimizes the sum of the squared errors (or residuals). This line minimizes the sum of the squared vertical distance between the observed values of y and those predicted by the line, yˆ (read “y-hat”). We represent this as “minimize∑residuals2”.

Question 20

Key Ideas about the Least-Squares Regression Line

Answer 20

The least-squares regression line, yˆ=b1x+b0, always contains the point (x-bar, y-bar).

Because sy and sx must both be positive, the sign of the linear correlation coefficient, r, and the sign of the slope of the least-squares regression line, b1, are the same.

The predicted value of y,yˆ, is an estimate of the mean value of the response variable for any value of the explanatory variable.

Question 21

Rounding the Slope and y-Intercept

Answer 21

Throughout the course, we agree to round the slope and y-intercept to four decimal places.

Question 22

Residual Plot

Answer 22

A scatter plot where the explanatory variable is plotted on the horizontal axis and the corresponding residual is on the vertical axis.

Question 23

What does it mean if the residual plot shows a discernable pattern?

Answer 23

The response and explanatory variables may not be linearly related.

Question 24

Univariate Data

Answer 24

Data in which a single variable was measured for each individual in the study.

Question 25

Bivariate Data

Answer 25

Data in which two variables are measured on an individual.

Question 26

Response Variable

Answer 26

The variable whose value is can be explained by the value of the explanatory or predictor variable.

Question 27

Explanatory Variable

Answer 27

The variable whose value is thought to play a role in determining the value of the response variable.

Question 28

Predictor Variable

Answer 28

Another name for the explanatory variable.

Question 29

Scatter Diagram

Answer 29

A graph that shows the relationship between two quantitative variables measured on the same individual. The explanatory variable is plotted on the horizontal axis, and the response variable is plotted on the vertical axis.

Question 30

Outside The Scope of the Model

Answer 30

The practice of using the regression model to make predictions for values of the explanatory variable that are much larger or much smaller than those observed.

Question 31

Deviation

Answer 31

Differences between the value of two quantitative variables.

Question 32

Total Deviation

Answer 32

The deviation between the observed and mean values of the response variable, denoted y - y-bar.

Question 33

Constant Error Variance

Answer 33

A requirement of the least-squares regression model that states the spread of the residuals should remain fairly constant when plotted against the explanatory variable.

Question 34

Outlier

Answer 34

An observation that is inconsistent with the overall pattern of the data.