Practice Problems on Correlation & Simple Regression
1. Suppose that, across a sample of stores, the correlation coefficient between beer prices and beer
sales is -0.65. What does this number indicate?
(a)
(b)
(c)
(d)
There is almost no variability in beer sales that is unexplained by beer price.
More beer sales tend to go along with lower beer prices.
As price increases by $1, beer sales decrease by 65%
All of the above are true.
2. The purpose of a scatterplot is:
(a)
(b)
(c)
(d)
To test for the significance of association in bivariate data.
To calculate the correlation coefficient.
To provide a visual picture of the relationship in bivariate data.
To determine a confidence interval for the regression slope.
3. The standard error of the sample regression slope tells you:
(a)
(b)
(c)
(d)
Approximately how different the slope coefficient will be in different samples.
Approximately how large the prediction errors are.
Approximately how spread out the Y scores are.
Approximately how much of the variability of Y is explained by X.
4. The correlation coefficient describes the _________ between 2 variables.
(a)
(b)
(c)
(d)
strength of curved association
strength of random association
strength of linear association
American Marketing Association
5. R2 is a measure used to describe the overall fit of the regression line. Which of the following
statements is/are correct about R2?
(a) In general, the closer the R2 is to 1, the better the fit of the regression line to the points in the
scatterplot.
(b) R2 tells you the proportion of the points in the scatterplot that fall right on the regression line.
(c) R2 will always decrease as you add new observations to your regression.
(d) All of the above are true statements about R2.
6. A cost accountant is developing a regression model to predict the total cost of producing a
batch of circuit boards as a function of the batch size. The independent and dependent variables
for this regression would be:
(a)
(b)
(c)
(d)
IV: circuit board
IV: batch size
IV: average cost
IV: total cost
DV: batch size
DV: total cost
DV: circuit board
DV: average cost
(The next 9 questions are based on the following information.)
Pete Estrian is looking to buy a used Honda Civic. He checks the Internet and finds a huge list of
Civics for sale in his area. He selects a random sample of 10 cars, ranging in age from 2 years
old to 15 years old. For each car, he enters the age (in years) and the offered sales price (in
thousands) into Excel. He runs a regression predicting price from age, and gets the following
(edited) output:
ANOVA
Regression
Residual
Total
Intercept
Age
df
1
8
9
SS
93.5
6.4
99.9
Coefficients Standard Error
12.10
0.60
-0.80
0.07
MS
93.51
0.80
F
117.0
t Stat
20.2
-10.8
P-value
0.00000004
0.000005
Significance F
0.000005
7. What is the equation for the regression line?
(a)
(b)
(c)
(d)
Predicted price = $12,100 - $800 * Age
Predicted price = $12,100 - $600 * Age
Predicted price = $20,200 - $10,800 * Age
Predicted price = $12,100 - $70 * Age
8. Car #5 in the sample was 10 years old and cost $4,000. Determine the predicted price and the
residual for this car.
(a)
(b)
(c)
(d)
Predicted price = $11,300;
Predicted price = $11,300;
Predicted price = $4,100;
Predicted price = $4,100;
residual = -$7,300
residual = $7,300
residual = $100
residual = -$100
9. Construct a 95% confidence interval for the drop in price associated with an additional year of
age.
(a)
(b)
(c)
(d)
($639, $961)
($667, $933)
($749, $851)
Cannot be determined from the information given
10. What is the correlation between Age and Price?
(a)
(b)
(c)
(d)
r= 0.97
r= -0.97
r= 0.80
r= -0.80
(Honda Civics Prices and Ages, continued.)
11. What is the typical difference between the predicted prices (based on the regression line) and
the actual prices for these cars?
(a)
(b)
(c)
(d)
about $70
about $600
about $890
about $2,530
12. What does the p-value of 0.000005 tell us?
(a) It is not very plausible that the population regression line relating Price to Age is flat.
(b) There is strong evidence that the slope of the population regression line is not 0.
(c) There is strong evidence that the knowing a Civic’s age would improves our prediction of its
price.
(d) All of the statements above are implied by the low p-value.
13. The average age of the cars in the sample is 7.1 years. What is the average price of the cars in
the sample?
(a)
(b)
(c)
(d)
$5,000
$6,420
$7,190
Cannot be determined from the information given.
14. What is the best conclusion we can draw about Honda Civics that are 5 years old, based on
the information we have?
(a) We conclude that the average price of 5-year old Civics is about $8,100, but we expect to see
some differences in prices for different 5-year Civics.
(b) We conclude that that all 5-year-old Civics should cost the same, about $8,100.
(c) We conclude that all 5-year-old Civics should cost more than all 6-year-old Civics, although
we can’t be completely sure by how much.
(d) All of the above are equally valid conclusions.
15. Suppose instead that Pete had taken a second sample, consisting of 6 cars that ranged in age
from 4 to 8 years old, and suppose that he regressed Price on Age for this second sample. How
would the standard error of the regression slope be different for this second sample (compared to
the first sample described on the previous page)?
(a) The standard error of the regression slope would probably be larger for the second sample.
(b) The standard error of the regression slope would probably be about the same for both
samples.
(c) The standard error of the regression slope would probably be smaller for the second sample.
(The next 5 questions deal with the following information.)
Below is partial output from a regression predicting consumption of beef (called
BeefConsumption, and measured in pounds of beef per person annually) from the price of beef
(called BeefPrice, and measured in cents per pound). [The data are from the United States from
1925 to 1941. During this period, the price of beef ranged from about 55 to 80 cents per pound.]
ANOVA
Regression
Residual
Total
df
1
15
16
SS
166.0
127.2
293.1
Intercept
BeefPrice
Coefficients
85.24
-0.47
Standard Error
7.30
0.11
MS
166.0
8.5
F
19.6
t Stat P-value
11.67 6.3E-09
-4.42 0.0005
Significance F
0.0005
Lower 95%
69.68
-0.69
Upper 95%
100.80
-0.24
16. In 1941, beef cost 56 cents per pound, and annual consumption of beef was 60.0 pounds per
person. Determine the predicted consumption of beef in 1941, and say whether the data point for
1941 is above or below the regression line.
(a) Predicted consumption of beef = 57.0;
data point is above the regression line
(b) Predicted consumption of beef = 57.0;
data point is below the regression line
(c) Predicted consumption of beef = 58.9;
data point is above the regression line
(d) Predicted consumption of beef = 58.9;
data point is below the regression line
17. What is the best interpretation of the y-intercept (85.24) for this regression?
(a) When beef was free in 1941, everyone consumed about 85 pounds of beef per year.
(b) The intercept doesn’t tell us much, because 0 isn’t a reasonable value to plug in for the price
of beef.
(c) The intercept doesn’t tell us much, because the p-value is too low.
(d) Both (a) and (c).
18. Determine the correlation between BeefPrice and BeefConsumption.
(a) r = +0.75
(b) r = -0.75
(c) r = +0.47
(d) r = -0.47
19. Determine the standard deviation of the residuals for this regression
(a) 2.9 pounds/person
(b) 0.11 pounds/person
(c) 8.5 pounds/person
(d) 0.75 pounds/person
20. Can we reject the null hypothesis that BeefConsumption is unrelated to BeefPrice? (use
alpha=.05)
(a) Yes, because the p-value for the slope is very small (p=.0005)
(b) Yes, because the 95% confidence interval for the slope does not include 0.
(c) No, because the t-statistic for the slope is negative (t=-4.42)
(d) Both (a) and (b).
21. Let X= the dosage of a stimulant (in milligrams) given to a patient, and let Y= the pulse rate
of the patient (in beats per minute). Data is gathered for 100 patients, and the resulting regression
equation is: predicted Y = 70 + 0.35 * X.
What is the best interpretation of the number 0.35?
(a) 35% of the variability in pulse rate is explained by the amount of stimulant.
(b) The correlation between amount of stimulant and pulse rate is 0.35.
(c) When a patient is given one additional milligram of stimulant, their pulse tends to increase
by about 0.35 beats per minute.
(d) When a patient is given 0.35 additional milligrams of stimulant, their pulse tends to increase
by about 1 beats per minute.
(The next 2 questions are based on the following information.)
22. Across a sample of data from New York City (one data point for each month), the correlation
between average monthly temperature (X) and number of homicides per month (Y) is observed to
be .40. The most appropriate interpretation is:
(a)
(b)
(c)
(d)
As the temperature increases by 10 degrees, we expect to see 4 more homicides.
For every additional 10 homicides, we expect temperature to increase by 4 degrees.
Average temperature explains 16% of the variability in number of homicides per month.
Average temperature explains 40% of the variability in number of homicides per month.
23. If the standard deviation of the monthly temperature measurements is 15 degrees, and the
standard deviation of number of homicides per month is 5, what is the slope of the regression line
predicting number of homicides from monthly temperature? (calculate directly)
24. Which of the following is not an assumption of the simple linear regression model?
(a)
(b)
(c)
(d)
The population mean of Y (for each level of X) is linearly related to X.
The population variance of Y is the same for each level of X.
The population distribution of Y for each level of X follows the normal distribution.
The population correlation between X and Y is equal to 1.
(The next 3 questions are based on the following information.)
25. A manager wants to predict the cost of travel (Y) for salespeople based on the number of
days (X) spent on each trip. Based on a sample of data for trips ranging from 1 to 5 days, the
following regression line is estimated: Y-hat = 180 + 130 X
What would be the best prediction for the cost for a trip lasting 3 days?
(a)
(b)
(c)
(d)
$390
$570
$930
Cannot be determined; the regression line shouldn't be used to predict costs for trips lasting
that long.
26. What would be the best prediction for the cost for a trip lasting 10 days?
(a)
(b)
(c)
(d)
$1300
$1480
$3100
Cannot be determined; the regression line shouldn't be used to predict costs for trips lasting
that long.
27. Suppose the average length of the trips in the sample is 4 days. What is the average cost of
travel for these trips? (calculate directly)
________________________________________
28. The standard error of the slope of the regression line depends on which of the following
quantities?
(a)
(b)
(c)
(d)
the number of observations in the sample
the variance of the scores on the independent variable
the variance of the residuals
all of the above
29. There is a relationship between caffeine consumption and performance on early-morning
tests as follows: at very low and at very high levels of caffeine consumption, performance is
worse than performance at moderate levels of caffeine consumption. Which of the following is
the best estimate of the correlation coefficient that describes this relationship?
(a)
(b)
(c)
(d)
r = -0.70
r = 0.00
r = +0.70
r = +1.00
(The next 9 questions are based on the following information.)
Mark E. Ting, a market researcher, is analyzing household earnings and spending data. He is
interested in relating monthly household grocery spending (in dollars) to annual household
income (in thousands of dollars). He gathers data for a sample of 9 households. Below is partial
output from his analysis.
Mean
SD
Source
Regression
Residual
Total
Intercept
Income
Annual Income (thousands of $)
52.8
29.9
df
1
7
8
SS
25543
3112
28656
MS
25543
445
Monthly Grocery Spending ($)
137.2
59.8
F
57.4
Coefficients Standard Error t Stat
37.51
14.92
2.51
1.89
0.25
7.57
P-value
0.0401
0.0001
30. What is the equation for the estimated regression line?
(a)
(b)
(c)
(d)
Predicted monthly grocery spending = 37.51 + 1.89 Annual income
Predicted monthly grocery spending = 1.89 + 37.51 Annual income
Predicted monthly grocery spending = 14.92 + 0.25 Annual income
Predicted monthly grocery spending = 1.89 + 0.25 Annual income
31. One household in the sample has annual income of $50,000 and spends $125 monthly on
groceries. What is the residual for this household?
(a)
(b)
(c)
(d)
-7
7
125
37.5
32. What is the correlation between monthly grocery spending and annual income?
(a)
(b)
(c)
(d)
.891
.944
1.89
Cannot be determined from the information given
33. In the ANOVA table, the F-statistic is 57.4. This F value is the test statistic used in testing
what null hypothesis?
(a)
(b)
(c)
(d)
H0: The slope of the sample regression line is 0
H0: The intercept of the sample regression line is 0
H0: The slope of the population regression line is 0
H0: The intercept of the population regression line is 0
(Grocery spending and income problems, continued)
34. What is the sum of squared deviations between each household's actual grocery spending and
the predicted grocery spending based on the regression line?
(a)
(b)
(c)
(d)
25543
3112
28656
445
35. What is the standard deviation of the residuals?
(a)
(b)
(c)
(d)
21.1
29.9
59.8
Cannot be determined from the information given
36. Construct a 90% confidence interval for the expected additional monthly grocery spending
for each additional $1000 of annual income.
(a)
(b)
(c)
(d)
(0, 3.78)
(1.31, 2.47)
(1.42, 2.36)
(9.23, 65.7)
37. Mark is especially interested in estimating the mean grocery spending for three types of
households: those with annual income of $20,000, those with annual income of $50,000, and
those with annual income of $100,000. He constructs three 95% confidence intervals for mean
monthly grocery spending, one for each of the three sub-populations of households. Which of
these confidence intervals will be the narrowest (i.e., for which set of households will the
estimation of mean grocery spending be the most precise?)
(a)
(b)
(c)
(d)
The confidence interval for the $20,000 households will be narrowest.
The confidence interval for the $50,000 households will be narrowest.
The confidence interval for the $100,000 households will be narrowest.
All three confidence intervals will be equally wide.
38. The 95% confidence interval for the mean grocery spending for households with annual
income of $20,000 is calculated to be (50,101). What is the best interpretation of this interval?
(a) If a new household with income of $20,000 was observed, we would be 95% confident that
this household's actual grocery spending would fall in the interval.
(b) If we observed 100 new households with income of $20,000, about 95 of them would have
grocery spending falling in the interval.
(c) We are 95% confident that the mean grocery spending of all households with income of
$20,000 falls in this interval.
(d) About 95% of all households in the sample have annual incomes between $50,000 and
$101,000.