Regression
BPS chapter 5
© 2006 W.H. Freeman and Company
Sum of squared errors
Which least-squares regression line would have a smaller sum of
squared errors (SSE)?
a)
b)
c)
The line in Plot A.
The line in Plot B.
It would be the same for both plots.
Sum of squared errors (answer)
Which least-squares regression line would have a smaller sum of
squared errors (SSE)?
a)
b)
c)
The line in Plot A.
The line in Plot B.
It would be the same for both plots.
Scatterplots
Look at the following scatterplot. What could we say about the
relationship between r and the slope of the regression line?
a)
b)
c)
d)
Since a is negative, r must also be negative.
Since b is negative, r must also be negative.
Since a is positive, r must also be positive.
Since b is positive, r must also be positive.
Scatterplots (answer)
Look at the following scatterplot. What could we say about the
relationship between r and the slope of the regression line?
a)
b)
c)
d)
Since a is negative, r must also be negative.
Since b is negative, r must also be negative.
Since a is positive, r must also be positive.
Since b is positive, r must also be positive.
Slope
Look at the following scatterplot. What would be a correct interpretation of the
slope?
a)
b)
c)
d)
As we increase our CO content by 1 mg, we increase the tar content by
1.01 mg.
As we increase our CO content by 0.66 mg, we increase the tar content by
1.01 mg.
As we increase our CO content by 0.66 mg, we increase the tar content by
0.66 mg.
As we increase our CO content by 1 mg, we increase the tar content by
0.66 mg.
Slope (answer)
Look at the following scatterplot. What would be a correct interpretation of the
slope?
a)
b)
c)
d)
As we increase our CO content by 1 mg, we increase the tar content
by 1.01 mg.
As we increase our CO content by 0.66 mg, we increase the tar content by
1.01 mg.
As we increase our CO content by 0.66 mg, we increase the tar content by
0.66 mg.
As we increase our CO content by 1 mg, we increase the tar content by
0.66 mg.
Regression line
Look at the following least-squares regression line. If a person
increased his/her weight by 10 pounds, by how much (in inches)
would one expect to see their waist girth increase?
a)
b)
c)
d)
0.1332
1.332
99.994
1.332 + 9.9994
Regression line (answer)
Look at the following least-squares regression line. If a person
increased his/her weight by 10 pounds, by how much (in inches)
would one expect to see their waist girth increase?
a)
b)
c)
d)
0.1332
1.332
99.994
1.332 + 9.9994
Regression line
Look at the following least-squares regression line. The Y-intercept
tells us the predicted waist girth for someone weighing how many
pounds?
a)
b)
c)
d)
0
0.1332
9.9994
Cannot be determined from the graph.
Regression line (answer)
Look at the following least-squares regression line. The Y-intercept
tells us the predicted waist girth for someone weighing how many
pounds?
a)
b)
c)
d)
0
0.1332
9.9994
Cannot be determined from the graph.
Residuals
Look at the following least-squares regression line. Compare the
squared errors (residuals) from the two Points A and B.
a)
b)
c)
d)
Point A’s would be greater than Point B’s.
Point A’s would be less than Point B’s.
Point A’s would be equal to Point B’s.
There is not enough information.
Residuals (answer)
Look at the following least-squares regression line. Compare the
squared errors (residuals) from the two Points A and B.
a)
b)
c)
d)
Point A’s would be greater than Point B’s.
Point A’s would be less than Point B’s.
Point A’s would be equal to Point B’s.
There is not enough information.
Percent of variation in Y
What percent of the variation in the sisters’ heights can be explained by
the heights of the brothers?
a)
b)
c)
d)
25.64%
(0.558)2 = 31.14%
52.7%
55.8%
Percent of variation in Y (answer)
What percent of the variation in the sisters’ heights can be explained by
the heights of the brothers?
a)
b)
c)
d)
25.64%
(0.558)2 = 31.14%
52.7%
55.8%
Correlation
The correlation between math SAT score and total SAT score is about r
= 0.9935. What is a correct conclusion that could be made?
a)
b)
c)
d)
The least-squares regression line of Y on X would have slope =
0.9935.
Math SAT scores explain about 98.7% (which is 0.99352) of the
variation in the total SAT scores.
About 99.35% of the time math SAT scores will accurately predict
total SAT scores.
Total SAT score is made up of 99.35% of the math SAT score.
Correlation (answer)
The correlation between math SAT score and total SAT score is about r
= 0.9935. What is a correct conclusion that could be made?
a)
b)
c)
d)
The least-squares regression line of Y on X would have slope =
0.9935.
Math SAT scores explain about 98.7% (which is 0.99352) of the
variation in the total SAT scores.
About 99.35% of the time math SAT scores will accurately predict
total SAT scores.
Total SAT score is made up of 99.35% of the math SAT score.
Residuals
Residual equals
a)
b)
c)
d)
Residuals (answer)
Residual equals
a)
b)
c)
d)
Residual plots
Residual plots are used to
a)
b)
c)
d)
Examine the relationship between two variables.
Identify the mean and spread of the residuals.
Check for independence of observations.
Magnify violations of regression assumptions.
Residual plots (answer)
Residual plots are used to
a)
b)
c)
d)
Examine the relationship between two variables.
Identify the mean and spread of the residuals.
Check for independence of observations.
Magnify violations of regression assumptions.
Residual plots
The following are regression assumptions:
1.
The relationship between X and Y can be modeled with a straight line.
2.
The variation in the Y values does not depend on the value of X (constant
variance).
The residual plot shown below indicates the violation of which regression
assumption?
a)
b)
c)
1
2
Neither
Residual plots (answer)
The following are regression assumptions:
1.
The relationship between X and Y can be modeled with a straight line.
2.
The variation in the Y values does not depend on the value of X (constant
variance).
The residual plot shown below indicates the violation of which regression
assumption?
a)
b)
c)
1
2
Neither
Residual plots
The following are regression assumptions:
1.
The relationship between X and Y can be modeled with a straight line.
2.
The variation in the Y values does not depend on the value of X (constant
variance).
The residual plot shown below indicates the violation of which regression
assumption?
a)
b)
c)
1
2
Neither
Residual plots (answer)
The following are regression assumptions:
1.
The relationship between X and Y can be modeled with a straight line.
2.
The variation in the Y values does not depend on the value of X (constant
variance).
The residual plot shown below indicates the violation of which regression
assumption?
a)
b)
c)
1
2
Neither
Correlation or regression
Which of the following measures the direction and strength of the linear
association between X and Y?
a)
b)
Correlation
Regression
Correlation or regression (answer)
Which of the following measures the direction and strength of the linear
association between X and Y?
a)
b)
Correlation
Regression
Correlation or regression
Which of the following makes no distinction between explanatory and
response variables?
a)
b)
Correlation
Regression
Correlation or regression (answer)
Which of the following makes no distinction between explanatory and
response variables?
a)
b)
Correlation
Regression
Correlation or regression
Which of the following is used for prediction?
a)
b)
Correlation
Regression
Correlation or regression (answer)
Which of the following is used for prediction?
a)
b)
Correlation
Regression
Regression line
A regression line always passes through the point
a)
b)
c)
d)
Regression line (answer)
A regression line always passes through the point
a)
b)
c)
d)
Correlation and slope
Which of the following best describes the relationship between
correlation and slope?
a)
b)
c)
d)
The correlation of X and Y equals the slope of the regression line
modeling the relationship between X and Y.
When the correlation between X and Y is zero, the slope of the
regression line modeling the relationship between X and Y is
negative.
The sign of the correlation between X and Y is the same as the sign
of the slope of the regression line modeling the relationship between
X and Y.
The correlation between X and Y is not related to the slope of the
regression line modeling the relationship between X and Y.
Correlation and slope (answer)
Which of the following best describes the relationship between
correlation and slope?
a)
b)
c)
d)
The correlation of X and Y equals the slope of the regression line
modeling the relationship between X and Y.
When the correlation between X and Y is zero, the slope of the
regression line modeling the relationship between X and Y is
negative.
The sign of the correlation between X and Y is the same as the
sign of the slope of the regression line modeling the
relationship between X and Y.
The correlation between X and Y is not related to the slope of the
regression line modeling the relationship between X and Y.
Regression line
Which of the following best measures the strength of fit of a regression
line?
a)
b)
c)
Correlation coefficient, r.
Square of the correlation coefficient, r2.
Square root of the correlation coefficient, r .
Regression line (answer)
Which of the following best measures the strength of fit of a regression
line?
a)
b)
c)
Correlation coefficient, r.
Square of the correlation coefficient, r2.
Square root of the correlation coefficient, r .
Causation
Researchers interviewed a group of women with knee pain awaiting
knee replacement surgery. They also interviewed a group of
women from the same geographical area with no knee pain. These
researchers reported that wearing high-heeled shoes caused the
knee pain which required surgery. As a savvy consumer of
statistics, you conclude that:
a)
b)
Because this was only an observational study, the researchers
should not make claims that the knee pain was caused by high
heels.
Because the study was a valid experiment, the researchers were
valid in their claim about high heels causing pain.
Causation (answer)
Researchers interviewed a group of women with knee pain awaiting
knee replacement surgery. They also interviewed a group of
women from the same geographical area with no knee pain. These
researchers reported that wearing high-heeled shoes caused the
knee pain which required surgery. As a savvy consumer of
statistics, you conclude that:
a)
b)
Because this was only an observational study, the researchers
should not make claims that the knee pain was caused by high
heels.
Because the study was a valid experiment, the researchers were
valid in their claim about high heels causing pain.
Linear regression
The following graph indicates the presence of
a)
b)
c)
Extrapolation.
An influential observation.
A lurking variable.
Linear regression (answer)
The following graph indicates the presence of
a)
b)
c)
Extrapolation.
An influential observation.
A lurking variable.
Linear regression
The following graph shows the linear relationship between diamond
size and price for diamonds size 0.35 carats or less. Using this
relationship to predict the price of a diamond that is 1 carat is
considered
a)
b)
c)
Extrapolation.
An influential observation.
Prediction.
Linear regression (answer)
The following graph shows the linear relationship between diamond
size and price for diamonds size 0.35 carats or less. Using this
relationship to predict the price of a diamond that is 1 carat is
considered
a)
b)
c)
Extrapolation.
An influential observation.
Prediction.
Linear regression
The diamonds mentioned in the previous question were of the same cut
and clarity. If diamonds of different cuts have different relationship
between size and price, we would say that
a)
b)
c)
Type of cut is a lurking variable.
Type of cut is a confounded variable.
Type of cut should be ignored.
Linear regression (answer)
The diamonds mentioned in the previous question were of the same cut
and clarity. If diamonds of different cuts have different relationship
between size and price, we would say that
a)
b)
c)
Type of cut is a lurking variable.
Type of cut is a confounded variable.
Type of cut should be ignored.