Inference for Regression
BPS chapter 23
© 2010 W.H. Freeman and Company
1. Hypothesis tests
Researchers at The Ohio State University wanted to know if they could
use the number of beers consumed by a student to predict the
student’s blood alcohol content (BAC). The following scatter plot
shows the data. From the scatter plot it appears that there is ___
correlation between number of beers and BAC.
a)
b)
c)
d)
positive
negative
No
All of the above
1. Hypothesis tests
Researchers at The Ohio State University wanted to know if they could
use the number of beers consumed by a student to predict the
student’s blood alcohol content (BAC). The following scatterplot
shows the data. From the scatter plot it appears that there is ___
correlation between number of beers and BAC.
a)
b)
c)
d)
positive
negative
No
All of the above
2. Hypothesis tests
Researchers at The Ohio State University wanted to know if they could
use the number of beers consumed by a student to predict the
student’s blood alcohol content (BAC). In order to know if the
number of beers consumed was a good predictor of BAC, they
tested H0 :   0, Ha :   0. What can we conclude from the
following table?
a)
b)
c)
d)
Because the P-value is 0.3320, there is a significant linear
relationship between the number of beers consumed and BAC.
Because the P-value is 0.0000, there is a significant linear
relationship between the number of beers consumed and BAC.
Because the P-value is 0.3320, there is no significant linear
relationship between the number of beers consumed and BAC.
Because the P-value is 0.0000, there is no significant linear
relationship between the number of beers consumed and BAC.
2. Hypothesis tests (answer)
Researchers at The Ohio State University wanted to know if they could
use the number of beers consumed by a student to predict the
student’s blood alcohol content (BAC). In order to know if the
number of beers consumed was a good predictor of BAC, they
tested H0 :   0, Ha :   0. What can we conclude from the
following table?
a)
b)
c)
d)
Because the P-value is 0.3320, there is a significant linear
relationship between the number of beers consumed and BAC.
Because the P-value is 0.0000, there is a significant linear
relationship between the number of beers consumed and BAC.
Because the P-value is 0.3320, there is no significant linear
relationship between the number of beers consumed and BAC.
Because the P-value is 0.0000, there is no significant linear
relationship between the number of beers consumed and BAC.
3. Conclusions
An article in a newspaper said that students who major in subjects that
have higher expected incomes after graduation are more likely to be
married. This conclusion is:
a)
b)
c)
Correct because the data were collected in a scientific way.
Incorrect because the results are likely biased due to lurking
variables.
Not reliable because it does not sound plausible.
3. Conclusions (answer)
An article in a newspaper said that students who major in subjects that
have higher expected incomes after graduation are more likely to be
married. This conclusion is:
a)
b)
c)
Correct because the data were collected in a scientific way.
Incorrect because the results are likely biased due to lurking
variables.
Not reliable because it does not sound plausible.
4. Linear regression
Which point represents “a” in our least-squares regression equation?
a)
b)
c)
d)
Point Q
Point S
Point R
Point T
4. Linear regression (answer)
Which point represents “a” in our least-squares regression equation?
a)
b)
c)
d)
Point Q
Point S
Point R
Point T
5. Correlation
If two quantitative variables, X and Y, have a correlation coefficient r =
0.80, which graph could be a scatterplot of the two variables?
a)
b)
c)
Plot A
Plot B
Plot C
5. Correlation (answer)
If two quantitative variables, X and Y, have a correlation coefficient r =
0.80, which graph could be a scatterplot of the two variables?
a)
b)
c)
Plot A
Plot B
Plot C
6. Residual
The following scatterplot shows the number of gold medals earned by
countries in 1992 versus how many earned in 1996. Which of the
points would have the smallest residual?
a)
b)
c)
d)
Point A
Point B
Point C
Point D
6. Residual (answer)
The following scatterplot shows the number of gold medals earned by
countries in 1992 versus how many earned in 1996. Which of the
points would have the smallest residual?
a)
b)
c)
d)
Point A
Point B
Point C
Point D
8. Appropriate analysis
Edwin Hubble collected data on the distance a galaxy is from the earth
and the velocity with which it appears to be receding. If he wanted
to investigate if there was a linear relationship between the distance
and the velocity, what type of analysis did he perform?
a)
b)
c)
d)
Two-sample t-test on means
2 analysis on proportions
Linear regression analysis
Matched pairs experiment
8. Appropriate analysis (answer)
Edwin Hubble collected data on the distance a galaxy is from the earth
and the velocity with which it appears to be receding. If he wanted
to investigate if there was a linear relationship between the distance
and the velocity, what type of analysis did he perform?
a)
b)
c)
d)
Two-sample t-test on means
2 analysis on proportions
Linear regression analysis
Matched pairs experiment
9. Linear regression
Edwin Hubble collected data on the distance a galaxy is from the earth
and the velocity with which it appears to be receding. He used the
following model:  y     x where x represents the distance the
galaxy is from the earth (in megaparsecs) and  y represents the
mean velocity (in km/sec) for all galaxies at that distance. What
does  represent in this problem?
a)
b)
c)
d)
The average velocity for a galaxy that is extremely close to earth.
The average change in velocity for a one-megaparsec increase in
distance for those galaxies in the sample.
The average velocity for all galaxies in the universe.
The average change in velocity for a one-megaparsec increase in
distance of all galaxies.
9. Linear regression (answer)
Edwin Hubble collected data on the distance a galaxy is from the earth
and the velocity with which it appears to be receding. He used the
following model:  y     x where x represents the distance the
galaxy is from the earth (in megaparsecs) and  y represents the
mean velocity (in km/sec) for all galaxies at that distance. What
does  represent in this problem?
a)
b)
c)
d)
The average velocity for a galaxy that is extremely close to earth.
The average change in velocity for a one-megaparsec increase in
distance for those galaxies in the sample.
The average velocity for all galaxies in the universe.
The average change in velocity for a one-megaparsec increase
in distance of all galaxies.
10. Linear regression
Edwin Hubble collected data on the distance a galaxy is from the earth
and the velocity with which it appears to be receding. Summarizing
his data with a scatterplot and generating the least-squares
regression line gave the following table:
Based on the information in the table, what is the correct equation for
the least-squares regression line?
a)
b)
c)
d)
e)
10. Linear regression (answer)
Edwin Hubble collected data on the distance a galaxy is from the earth
and the velocity with which it appears to be receding. Summarizing
his data with a scatterplot and generating the least-squares
regression line gave the following table:
Based on the information in the table, what is the correct equation for
the least-squares regression line?
a)
b)
c)
d)
e)
11. Residuals
Edwin Hubble collected data on the distance a galaxy is from the earth and the
velocity with which it appears to be receding. By looking at the following
residual plot and histogram of the residuals, what conclusion should be
made about the conditions for performing the linear regression?
a)
b)
c)
d)
e)
Because the residual plot shows no pattern and the histogram is
approximately bell-shaped, the conditions are met.
The residual plot implies that the data violate the assumption of normality.
The histogram of the residuals shows that the data are extremely rightskewed.
Neither plot tells us anything about the assumptions for doing inference for
regression.
The residual plot implies that the data violate the assumption of linearity.
11. Residuals (answer)
Edwin Hubble collected data on the distance a galaxy is from the earth and the
velocity with which it appears to be receding. By looking at the following
residual plot and histogram of the residuals, what conclusion should be
made about the conditions for performing the linear regression?
a)
b)
c)
d)
e)
Because the residual plot shows no pattern and the histogram is
approximately bell-shaped, the conditions are met.
The residual plot implies that the data violate the assumption of normality.
The histogram of the residuals shows that the data are extremely rightskewed.
Neither plot tells us anything about the assumptions for doing inference for
regression.
The residual plot implies that the data violate the assumption of linearity.
12. Linear relationship
Edwin Hubble collected data on the distance a galaxy is from the earth
and the velocity with which it appears to be receding. If the
researchers want to test whether there is a positive linear
relationship between the distance and velocity, what hypotheses
could be used?
a)
b)
c)
d)
12. Linear relationship (answer)
Edwin Hubble collected data on the distance a galaxy is from the earth
and the velocity with which it appears to be receding. If the
researchers want to test whether there is a positive linear
relationship between the distance and velocity, what hypotheses
could be used?
a)
b)
c)
d)
13. Relationships
The following plot shows a person’s score on a sobriety test versus
their blood alcohol content. Which statement is NOT true about this
plot?
a)
b)
c)
d)
An outlier is present in the dataset.
A relationship exists between BAC and the test score.
The relationship could be modeled with a straight line.
There is a positive relationship between the two variables.
13. Relationships (answer)
The following plot shows a person’s score on a sobriety test versus
their blood alcohol content. Which statement is NOT true about this
plot?
a)
b)
c)
d)
An outlier is present in the dataset.
A relationship exists between BAC and the test score.
The relationship could be modeled with a straight line.
There is a positive relationship between the two variables.
14. Conclusions
The average height of people in the United States has been increasing
for decades. Similarly there is evidence that the number of plant
species is decreasing over these decades. An appropriate
conclusion to draw from these observations would be that
a)
b)
c)
Even though they appear to be associated, we could not conclude
association.
Growing adults are causing the number of plant species to
decrease.
There is a positive relationship between the two variables.
14. Conclusions (answer)
The average height of people in the United States has been increasing
for decades. Similarly there is evidence that the number of plant
species is decreasing over these decades. An appropriate
conclusion to draw from these observations would be that
a)
b)
c)
Even though they appear to be associated, we could not
conclude association.
Growing adults are causing the number of plant species to
decrease.
There is a positive relationship between the two variables.