Chapter 21—Simple linear regression and correlation
MULTIPLE CHOICE
1. Testing whether the slope of the population regression line could be zero is equivalent to testing
whether the:
A.
sample coefficient of correlation could be zero.
B.
standard error of estimate could be zero.
C.
population coefficient of correlation could be zero.
D.
sum of squares for error could be zero.
2. The regression line
y
ö
= 3 + 2x has been fitted to the data points (4,8), (2,5), and (1,2). The sum of the
squared residuals will be:
A.
7.
B.
15.
C.
8.
D.
22.
3. The following sums of squares are produced:
(yi−(y-bar))2 = 250, (yi−(yi-hat))2 = 100, ((yi-hat)−(y-bar))2 = 150
The percentage of the variation in y that is explained by the variation in x is:
A.
60% .
B.
75% .
C.
40% .
D.
50% .
4. In simple linear regression, most often we perform a two-tail test of the population slope
1
to
determine whether there is sufficient evidence to infer that a linear relationship exists. The null
hypothesis is stated as:
A.
0: 10 =
H
.
B.
110 :bH =
.
C.
rH =
10 :
.
D.
=
10 :
H
s
.
5. Given that ssx = 2500, ssy = 3750, ssxy = 500 and n = 6, the standard error of estimate is:
A.
12.247.
B.
24.933.
C.
30.2076.
D.
11.180.
6. Given the least squares regression line
y
ö
= 5 –2x:
A.
the relationship between x and y is positive.
B.
the relationship between x and y is negative.
C.
as x increases, so does y.
D.
as x decreases, so does y.
7. If an estimated regression line has a y-intercept of 13 and a slope of –2, then when x = 0, the actual
value of y is:
A.
11.
B.
-2.
C.
13.
D.
unknown.
8. The symbol for the population coefficient of correlation is:
A.
r.
B.
.
C.
r
2
.
D.
2
.
9. Given a specific value of x and confidence level, which of the following statements is correct?
A.
The confidence interval estimate of the expected value of y can be calculated but the
prediction interval of y for the given value of x cannot be calculated.
B.
The confidence interval estimate of the expected value of y will be wider than the
prediction interval.
C.
The prediction interval of y for the given value of x can be calculated but the confidence
interval estimate of the expected value of y cannot be calculated.
D.
The confidence interval estimate of the expected value of y will be narrower than the
prediction interval.
10. The symbol for the sample coefficient of correlation is:
A.
r.
B.
.
C.
2
r
.
D.
2
.
11. Given that the sum of squares for error is 50 and the sum of squares for regression is 140, the
coefficient of determination is:
A.
0.736.
B.
0.357.
C.
0.263.
D.
2.800.
12. A regression line using 25 observations produced SSR = 118.68 and SSE = 56.32. The standard error
of estimate was:
A.
2.1788.
B.
1.5648.
C.
1.5009.
D.
2.2716.
13. Given the least squares regression line y-hat = 3.52 – 1.27x, and a coefficient of determination of 0.81,
the coefficient of correlation is:
A.
–0.85.
B.
0.85.
C.
–0.90.
D.
0.90.
14. The Spearman rank correlation coefficient must be used to determine whether a relationship exits
between two variables when:
A.
one of the variables may be ordinal.
B.
both of the variables may be ordinal.
C.
both variables are interval and the normality requirement may not be satisfied.
D.
All of the above statements are correct.
15. Which value of the coefficient of correlation r indicates a stronger correlation than 0.75?
A.
0.55.
B.
–0.75.
C.
0.60.
D.
–0.85.
16. If the coefficient of correlation is 0.90, the percentage of the variation in the dependent variable y that
is explained by the variation in the independent variable x is:
A.
90%.
B.
81%.
C.
0.90%.
D.
0.81%.
17. If the coefficient of determination is 0.975, then the slope of the regression line:
A.
must be positive.
B.
must be negative.
C.
could be either positive or negative.
D.
None of the above answers if correct.
18. In regression analysis, if the coefficient of determination is 1.0, then:
A.
the sum of squares for error must be 1.0.
B.
the sum of squares for regression must be 1.0.
C.
the sum of squares for error must be 0.0.
D.
the sum of squares for regression must be 0.0.
19. Correlation analysis is used to determine:
A.
the strength of the relationship between x and y.
B.
the least squares estimates of the regression parameters.
C.
the predicted value of y for a given value of x.
D.
the coefficient of determination.
20. If the coefficient of correlation is –0.90, the percentage of the variation in y that is explained by the
variation in x is:
A.
90%.
B.
–90%.
C.
–81%.
D.
81%.
21. If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of
correlation must be:
A.
1.0.
B.
–1.0.
C.
either 1.0 or –1.0.
D.
0.0.
22. If the coefficient of correlation between x and y is close to 1.0, this indicates that:
A.
y causes x to happen.
B.
x causes y to happen.
C.
Both A and B are correct answers.
D.
There may or may not be any causal relationship between x and y.
23. A regression analysis between height y (in cm) and age x (in years) of 2 to 10 years old boys yielded
the least squares line y-hat = 87 + 6.5x. This implies that by each additional year height is expected to:
A.
increase by 93.5cm.
B.
increase by 6.5cm.
C.
increase by 87cm.
D.
decrease by 6.5cm.
24. A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line
y
ö
= 75 +6x. This implies that if $800 is spent on advertising, then the predicted amount of sales (in
dollars) is:
A.
$4875.
B.
$123 000.
C.
$487 500.
D.
$12 300.
25. If the coefficient of correlation is –0.30, then the coefficient of determination is:
A.
–0.09.
B.
–0.30.
C.
0.09.
D.
0.70.
26. A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line
y
ö
= 80 000 + 5x. This implies that an:
A.
increase of $1 in advertising is expected to result in an increase of $5 in sales.
B.
increase $5 in advertising is expected to result in an increase of $5000 in sales.
C.
increase of $1 in advertising is expected to result in an increase of $80 005 in sales.
D.
increase of $1 in advertising is expected to result in an increase of $5000 in sales.
27. In regression analysis, if the coefficient of correlation is –1.0, then:
A.
the sum of squares for error is –1.0.
B.
the sum of squares for regression is 1.0.
C.
the sum of squares for error and sum of squares for regression are equal.
D.
the sum of squares for regression and total variation in y are equal.
28. Given the data points (x,y) = (3,3), (4,4), (5,5), (6,6), (7,7), the least squares estimates of the
y-intercept and slope are respectively:
A.
0 and 1.
B.
–1 and 0.
C.
5 and 5.
D.
5 and 0.
29. Of the values of the coefficient of determination listed below, which one implies the greatest value of
the sum of squares for regression, given that the total variation in y is 1800?
A.
0.69.
B.
0.96.
C.
96.
D.
–100.
30. When all the actual values of y and the predicted values of y are equal, the standard error of estimate
will be:
A.
1.0.
B.
–1.0.
C.
0.0.
D.
2.0.
31.
32. In order to estimate with 95% confidence the expected value of y in a simple linear regression
problem, a random sample of 10 observations is taken. Which of the following t-table values listed
below would be used?
A.
2.228.
B.
2.306.
C.
1.860.
D.
1.812.
33. In testing the hypotheses:
0:
0=
s
H
0:
1
s
H
,
the Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test
statistic is:
A.
2.75.
B.
18.178.
C.
2.723.
D.
17.995.
34. Which of the following techniques is used to predict the value of one variable on the basis of other
variables?
A.
Correlation analysis.
B.
Coefficient of correlation.
C.
Covariance.
D.
Regression analysis.
35. Which of the following statistics and procedures can be used to determine whether a linear model
should be employed?
A.
The standard error of estimate.
B.
The coefficient of determination.
C.
The t-test of the slope.
D.
All of the above are correct answers.
36. The residual is defined as the difference between:
A.
the actual value of y and the estimated value of y.
B.
the actual value of x and the estimated value of x.
C.
the actual value of y and the estimated value of x.
D.
the actual value of x and the estimated value of y.
37. The standardised residual is defined as:
A.
residual divided by the standard error of estimate.
B.
residual multiplied by the square root of the standard error of estimate.
C.
residual divided by the square of the standard error of estimate.
D.
residual multiplied by the standard error of estimate.
38. The standard error of estimate,
s
, is given by:
A.
SSE/(n – 2).
B.
)2/( −nSSE
.
C.
)2/( −nSSE
.
D.
SSE/
2−n
.
39. If the standard error of estimate
s
= 15 and n = 12, then the sum of squares for error, SSE, is:
A.
150.
B.
180.
C.
225.
D.
2250.
40. The smallest value that the standard error of estimate
s
can assume is:
A.
–1.
B.
0.
C.
1.
D.
–2.
41. If cov(X,Y) = −350, sx2 = 900 and sy2 = 225, then the coefficient of determination is:
A.
0.8819.
B.
0.7778.
C.
−0.0017.
D.
0.0017.
42. The coefficient of determination
2
R
measures the amount of:
A.
variation in y that is explained by variation in x.
B.
variation in x that is explained by variation in y.
C.
variation in y that is unexplained by variation in x.
D.
variation in x that is unexplained by variation in y.
43. If we are interested in determining whether two variables are linearly related, it is necessary to:
A.
perform the t-test of the slope
1
.
B.
perform the t-test of the coefficient of correlation
.
C.
do either A or B since they are identical.
D.
calculate the standard error of estimate
s
.
44. In the first-order linear regression model, the population parameters of the y-intercept and the slope
are:
A.
0
b
and
1
b
.
B.
0
b
and
1
.
C.
0
and
1
b
.
D.
0
and
1
.
45. The standard error of estimate,
s
, is a measure of:
A.
variation of y around the regression line.
B.
variation of x around the regression line.
C.
variation of y around the mean
y
.
D.
variation of x around the mean
x
.
46. In the simple linear regression model, the y-intercept represents the:
A.
change in y per unit change in x.
B.
change in x per unit change in y.
C.
value of y when x = 0.
D.
value of x when y = 0.
47. In the first-order linear regression model, the population parameters of the y-intercept and the slope are
estimated by:
A.
0
b
and
1
b
.
B.
0
b
and
1
.
C.
0
and
1
b
.
D.
0
and
1
.
48. The least squares method requires that the variance
2
of the error variable is a constant no matter
what the value of x is. When this requirement is violated, the condition is called:
A.
multicollinearity.
B.
heteroscedasticity.
C.
homoscedasticity.
D.
autocorrelation.
49. In the simple linear regression model, the slope represents the:
A.
value of y when x = 0.
B.
change in y per unit change in x.
C.
value of x when y = 0.
D.
change in x per unit change in y.
50. When the sample size n is greater than 30, the Spearman rank correlation coefficient
s
r
is
approximately normally distributed with:
A.
mean 0 and standard deviation 1.
B.
mean 1 and standard deviation
1−n
.
C.
mean 1 and standard deviation 1/
1−n
.
D.
mean 0 and standard deviation 1/
1−n
.
51. The Pearson coefficient of correlation r equals 1 when there is/are no:
A.
explained variation.
B.
unexplained variation.
C.
y-intercept in the model.
D.
outliers.
52. In regression analysis, the residuals represent the:
A.
difference between the actual y values and their predicted values.
B.
difference between the actual x values and their predicted values.
C.
square root of the coefficient of determination.
D.
change in y per unit change in x.
53. In regression analysis, the coefficient of determination,
2
R
, measures the amount of variation in y that
is:
A.
caused by the variation in x.
B.
explained by the variation in x.
C.
unexplained by the variation in x.
D.
Both A and B are correct answers.
54. In a regression problem the following pairs (x,y) are given: (1,2), (2.5,2), (3,2), (5,2) and (5.3,2). This
indicates that the:
A.
correlation coefficient is 0.
B.
correlation coefficient is 1.
C.
correlation coefficient is −1.
D.
coefficient of determination is between 0 and 1.
55. In a regression problem, if the coefficient of determination is 0.95, this means that:
A.
95% of the y values are positive.
B.
95% of the variation in y can be explained by the variation in x.
C.
95% of the x values are equal.
D.
95% of the variation in x can be explained by the variation in y.
56. In a simple linear regression problem, the following statistics are calculated from a sample of 10
observations:
−− ))(( yyxx
= 2250,
x
s
= 10,
x
= 50,
y
= 75
The least squares estimates of the slope and y-intercept are respectively:
A.
225 and −1117.5.
B.
2.5 and −5.
C.
25 and −117.5.
D.
25 and 117.5.
57. Which of the following is not a required condition for the error variable
in the simple linear
regression model?
A.
The probability distribution of
is normal.
B.
The mean of the probability distribution of
is zero.
C.
The standard deviation
of
is a constant, no matter what the value of x.
D.
The values of
are auto correlated.
58. When the variance,
2
, of the error variable
is a constant no matter what the value of x is, this
condition is called:
A.
homocausality.
B.
heteroscedasticity.
C.
homoscedasticity.
D.
heterocausality.
59. In simple linear regression, the coefficient of correlation r and the least squares estimate
1
b
of the
population slope
1
:
A.
must be equal.
B.
must have opposite signs.
C.
must have the same sign.
D.
may have opposite signs or the same sign.
60. If a simple linear regression model has no y-intercept, then:
A.
when x = 0, so does y.
B.
y is always zero.
C.
when y = 0, so does x.
D.
y is always equal to x.
61. The least squares method for determining the best fit minimises:
A.
total variation in the dependent variable.
B.
the sum of squares for error.
C.
the sum of squares for regression.
D.
All of the above are correct answers.
62. On the least squares regression line y-hat= 2 – 3x, the predicted value of y equals:
A.
–1.0 when x = –1.0.
B.
1.0 when x = 1.0.
C.
5.0 when x = –1.0.
D.
5.0 when x = 1.0.
63. In simple linear regression, which of the following statements indicates no linear relationship between
the variables x and y?
A.
The coefficient of determination is 1.0.
B.
The coefficient of correlation is 0.0.
C.
The sum of squares for error is 0.0.
D.
The sum of squares for regression is relatively large.
64. If the sum of squared residuals is zero, then the:
A.
coefficient of determination must be 1.0.
B.
coefficient of correlation must be 1.0.
C.
coefficient of determination must be 0.0.
D.
coefficient of correlation must be 0.0.
65. In a regression problem, if all the values of the independent variable are equal, then the coefficient of
determination must be:
A.
1.0.
B.
0.5.
C.
0.0.
D.
–1.0.
TRUE/FALSE
1. If the value of the sum of squares for error, SSE, equals zero, then the coefficient of determination
must equal zero.
2. If all the values of an independent variable x are equal, then regressing a dependent variable y on x will
result in a coefficient of determination of zero.
3. When the actual values y of a dependent variable and the corresponding predicted values
ö
y
are the
same, the standard error of the estimate will be 1.0.
4. The value of the sum of squares for regression, SSR, can never be smaller than 0.0.
5. The value of the sum of squares for regression, SSR, can never be smaller than 1.
6. A direct relationship between an independent variable x and a dependent variably y means that x and y
move in the opposite directions.
7. A direct relationship between an independent variable x and a dependent variably y means that the
variables x and y increase or decrease together.
8. The variance of the error variable,
, is required to be constant. When this requirement is satisfied,
the condition is called homoscedasticity.
9. The variance of the error variable,
2
, is required to be constant. When this requirement is violated,
the condition is called heteroscedasticity.
10. The method of least squares requires that the sum of the squared deviations between actual y values in
the scatter diagram and y values predicted by the regression line be maximised.
11. In a simple linear regression model, testing whether the slope,
1
, of the population regression line is
zero is the same as testing whether or not the population coefficient of correlation,
, equals one.
12. In developing a 95% confidence interval for the expected value of y from a simple linear regression
problem involving a sample of size 10, the appropriate table value would be 2.306.
13. When the actual values y of a dependent variable and the corresponding predicted values
ö
y
are the
same, the standard error of estimate,
s
, will be 0.0.
14. In developing a 90% confidence interval for the expected value of y from a simple linear regression
problem involving a sample of size 15, the appropriate table value would be 1.761.
15. Regardless of the value of x, the standard deviation of the distribution of y values about the regression
line is supposed to be constant. This assumption of equal standard deviations about the regression line
is called multicollinearity.
16. Another name for the residual term in a regression equation is random error.
17. A simple linear regression equation is given by
5.25 3.8
ö
y x= +
. The point estimate of
y
when
x
= 4
is 20.45.
18. The vertical spread of the data points about the regression line is measured by the y-intercept.
2
19. If there is no linear relationship between two variables
x
and
y
, the coefficient of determination must
be ±1.0.
20. In simple linear regression, the divisor of the standard error of estimate,
s
, is n – 2.
21. The value of the sum of squares for regression, SSR, can never be larger than the value of sum of
squares for error, SSE.
22. In order to estimate with 95% confidence a particular value of
y
for a given value of
x
in a simple
linear regression problem, a random sample of 20 observations is taken. The appropriate table value
that would be used is 2.101.
23. A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line
y
ö
= 60 + 5x. This implies that an increase of $1 in advertising is expected to result in an increase of
$65 in sales.
24. A regression analysis between weight
y
(in kilograms) and height
x
(in centimetres) yielded the least
squares line
y
ö
= 135 + 6
x
. This implies that if the height is increased by 1 cm, the weight is expected
to increase on average by 6 kilograms.
25. A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line
y-hat = 77 +8x. This implies that if advertising is $600, then the predicted amount of sales (in dollars)
is $4877.
26. When the actual values y of a dependent variable and the corresponding predicted values
ö
y
are the
same, the standard error of estimate,
s
, will be –1.0.
27. In a simple linear regression problem, the least squares line is
y
ö
= –3.75 + 1.25
x
, and the coefficient
of determination is 0.81. The coefficient of correlation must be –0.90.
28. In a regression problem the following pairs (x, y) are given: (3,–2), (3,–1), (3,0), (3,1) and (3,2). This
indicates that the coefficient of correlation is –1.
29. The residual
i
r
is defined as the difference between the actual value
i
y
and the estimated value
ö
i
y
.
30. The value of the sum of squares for regression, SSR, can never be equal to the value of total sum of
squares, SST.
31. We standardise residuals in the same way that we standardise all variables, by subtracting the mean
and dividing by the variance.
32. In regression analysis, if the coefficient of determination is 1.0, then the coefficient of correlation must
be 1.0 or −1.0.
33. Correlation analysis is used to determine the strength of the relationship between an independent
variable x and a dependent variable y.
34. In a simple linear regression problem, if the coefficient of determination is 0.95, this means that 95%
of the variation in the independent variable x can be explained by the regression line.
35. The regression line
y
ö
= 2 + 3x has been fitted to the data points (4,11), (2,7), and (1,5). The residual
sum of squares will be 10.0.
36. If the coefficient of correlation is –0.7, then the percentage of the variation in y that is explained by the
regression line is 49%.
37. If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of
correlation must be 1.0.
38. The residuals are observations of the error variable
. Consequently, the minimised sum of squared
deviations is called the sum of squares for error, denoted SSE.
39. If the standard error of estimate
s
= 20 and n = 8, then the sum of squares for error, SSE, is 2400.
40. Statisticians have shown that the sample y-intercept
0
b
and sample slope coefficient
1
b
are unbiased
estimators of the population regression parameters
0
and
1
.
41. The probability distribution of the error variable
is supposed to be normal, with mean E(
) = 0 and
constant standard deviation
.
42. Given that cov(x,y) = 8,
2
y
s
= 14,
2
x
s
= 10 and n = 6, the value of the sum of squares for error, SSE, is
38.
43. Given that cov(x,y) = 10,
2
y
s
= 15,
2
x
s
= 8 and n = 12, the value of the standard error of estimate,
s
, is
2.75.
44. If the error variable
is normally distributed, the test statistic for testing
0 1
: 0H
=
in a simple
linear regression follows the Student t–distribution with n – 1 degrees of freedom.
45. Given that cov(x,y) = 8.5,
2
y
s
= 8 and
2
x
s
= 10, the value of the coefficient of determination is 0.95.
46. The coefficient of determination is the coefficient of correlation squared. That is,
2 2
R r=
.
47. Except for the values r = –1, 0 and 1, we cannot be specific in our interpretation of the coefficient of
correlation r. However, when we square it, we produce a more meaningful statistic.
48. Given that SSE = 150 and SSR = 450, the proportion of the variation in y that is explained by the
variation in x is 0.75.
49. Given that SSE = 84 and SSR = 358.12, the coefficient of correlation (also called the Pearson
coefficient of correlation) must be 0.90.
50. The confidence interval estimate of the expected value of y will be narrower than the prediction
interval for the same given value of x and confidence level. This is because there is less error in
estimating a mean value than in predicting an individual value.
51. An outlier is an observation that is unusually small or unusually large.
52. One method of diagnosing autocorrelation is to plot the residuals against the time periods to see
whether some pattern emerges.
SHORT ANSWER
1. An economist wanted to analyse the relationship between the speed of a car (x) in kilometres per hour
(kmph) and its fuel consumption (y) in kilometres per litre (kmpl). In an experiment, a car was
operated at several different speeds and for each speed the fuel consumption was measured. The data
obtained are shown below.
Speed (mph)
25
35
45
50
60
65
70
Fuel consumtion
(mpg)
40
39
37
33
30
27
25
a. Find the least squares regression line.
b. Calculate the standard error of estimate, and describe what this statistic tells you about the
regression line.
c. Do these data provide sufficient evidence at the 5% significance level to infer that a linear
relationship exists between higher speeds and lower fuel consumption?
d. Predict with 99% confidence the fuel consumption of a car traveling at 55 kmph.
2. A television rating wants to determine whether married couples tend to agree about the quality of the
television shows they watch. Ten couples are asked to rate a particular comedy series on a 7-point
scale where 1 = terrible and 7 = excellent. The results are shown below.
Husband’s rating
3
6
6
5
4
5
7
4
5
5
Wife’s rating
5
5
4
5
4
4
6
3
4
5
Do these data provide sufficient evidence at the 5% significance level to conclude that the husband’s
and the wife’s ratings are positively related?
3. The following 10 observations of variables x and y were collected.
x
1
2
3
4
5
6
7
8
9
10
y
36
29
29
26
23
20
12
11
8
3
a. Find the least squares regression line.
b. Calculate the standard error of estimate.
c. Test to determine if there is enough evidence at the 5% significance level to indicate that x and y
are negatively linearly related.
d. Calculate the coefficient of correlation, and describe what this statistic tells you about the
regression line.
4. The dean of a faculty of business in Victoria believes that students who do well in ‘soft’ courses like
organisational behaviour do poorly in ‘hard’ courses like business statistics. In order to test his belief,
he takes a random sample of 10 students and records their test grades in organisational behaviour and
statistics. The results are shown below. Do these data provide sufficient evidence at the 5%
significance level to support the dean’s claim?
Student
Organisational
Behaviour Grade
Business Statistics
Grade
1
C
A
2
D
A
3
A
C
4
B
C
5
A
D
6
C
B
7
B
C
8
A
C
9
C
A
10
B
C
5. A scatter diagram includes the following data points:
x
3
2
5
4
5
y
9
6
11
11
15
Two regression models are proposed:
Model 1: y-hat = 1.85 + 2.40x.
Model 2: y-hat = 1.79 + 2.54x.
Using the least squares method, which of these regression models provides the better fit to the data?
Why?
6. A consultant for a beer company wanted to determine whether those who drink a lot of beer actually
enjoy the taste more than those who drink moderately or rarely. She took a random sample of eight
men and asked each how many beers they typically drink per week. She also asked them to rate their
favourite brand of beer on a 10-point scale (1 = bad, 10 = excellent). The results are shown below. Can
we infer at the 5% significance level that frequent beer drinkers rate their favourite beer more highly
than less frequent drinkers?
Beer
drinker
Typical weekly
consumption
Rating
1
4
6
2
3
6
3
12
9
4
15
8
5
7
8
6
9
6
7
1
5
8
10
8
7. Consider the following data values of variables x and y.
x
3
5
7
9
11
14
y
7
10
17
20
27
35
Determine the least squares regression line, and find the predicted value of y for x = 7.
8. Consider the following data values of variables x and y.
x
3
5
7
9
11
14
y
7
10
17
20
27
35
What does the value of the slope of the regression line tell you?
9. Consider the following data values of variables x and y.
x
3
5
7
9
11
14
y
7
10
17
20
27
35