CHAPTER
Correlation and Regression
9
9.1 CORRELATION
9.1 Try It Yourself Solutions
1ab.
2ab.
c. No, it appears that there is no linear correlation between height and pulse rate.
4. (a)
x y xy x2 y2
1 12.5 12.5 1 156.25
10 8.7 87.0 100 75.69
5 14.6 73.0 25 213.16
354 CHAPTER 9 │ CORREALTION AND REGRESSION
(b)
()()
() ()
22
22
nxy x y
r
nx x ny y
∑−∑ ∑
=
∑−∑ ∑−∑
5ab. 0.750r≈
c. Because r is close to 1, this suggests a strong positive linear correlation between the salaries
and the average attendances at home games.
7a. 0: 0; : 0
a
HHρρ=≠
b. 0.01α=
c. d.f. = 228n−=
d. 02.763;t=± Rejection regions: 2.763 or 2.763tt<− >
9.1 EXERCISE SOLUTIONS
1. Increase
2. Decrease
CHAPTER 9 │ CORRELATION AND REGRESSION 355
8. Answers will vary. Sample answer: The fact that two variables have a linear relationship does not
necessarily imply that one variable is the cause of the other.
9. Negative linear correlation
10. No linear correlation
14. Perfect positive linear correlation
15. (c), You would expect a positive linear correlation between age and income.
16. (d), You would not expect age and height to be correlated.
21. (a)
356 CHAPTER 9 │ CORREALTION AND REGRESSION
(b)
x y xy x2 y2
16 109 1744 256 11,881
25 122 3050 625 14,884
39 143 5577 1521 20,449
()()
() ()
()()()
()()
()()
22
22
22
10 68,173 416 1512
10 20,398 416 10 239,514 1512
nxy x y
r
nx x ny y
∑−∑ ∑
=
∑−∑ ∑−∑
−
=
−−
22. (a)
(b)
x y xy x2 y2
1 3 3 1 9
2 400 880 4 193,600
3 1200 3600 9 1,440,000
4 1500 6000 16 2,250,000
CHAPTER 9 │ CORRELATION AND REGRESSION 357
23. (a)
(b)
x y xy x2 y2
0 40 0 0 1600
1 41 41 1 1681
2 51 102 4 2601
4 48 192 16 2304
4 64 256 16 4096
5 69 345 25 4761
()()
() ()
()()()
()() ()()
22
22
22
13 4620 60 891
13 346 60 13 65, 451 891
nxy x y
r
nx x ny y
∑−∑ ∑
=
∑−∑ ∑−∑
−
=
−−
358 CHAPTER 9 │ CORREALTION AND REGRESSION
24. (a)
(b)
x y xy x2 y2
0 96 0 0 9216
1 85 85 1 7225
2 82 164 4 6724
3 74 222 9 5476
3 95 285 9 9025
5 68 340 25 4624
5 76 380 25 5776
(c) Strong negative linear correlation
25. (a)
CHAPTER 9 │ CORRELATION AND REGRESSION 359
(b)
x y xy x2 y2
300 961 288,300 90,000 923,521
258 891 229,878 66,564 793,881
250 937 234,250 62,500 877,969
()()
() ()
()()()
()()
()()
22
22
22
8 1,359,512 1912 5562
8 463,362 1912 8 4,258,086 5562
nxy x y
r
nx x ny y
∑−∑ ∑
=
∑−∑ ∑−∑
−
=
−−
26. (a)
(b)
x y xy x2 y2
0 1116.3 0.000 0 6
1.25 10×
5 1096.9 5484.5 25 6
1.2 10×
10 1077.3 10,773 100 6
1.16 10×
15 1057.2 15,858 225 6
1.12 10×
20 1036.8 20,736 400 6
1.07 10×
360 CHAPTER 9 │ CORREALTION AND REGRESSION
()()
() ()
22
22
nxy x y
r
nx x ny y
∑−∑ ∑
=
∑−∑ ∑−∑
27. (a)
(b)
x y xy x2 y2
6.00 2.45 14.7000 36.0000 6.0025
1.44 0.15 0.2160 2.0736 0.0225
4.44 0.62 2.7528 19.7136 0.3844
3.38 0.91 3.0758 11.4244 0.8281
()()
() ()
()()()
()()
()()
22
22
22
12 42.8364 42.23 9.63
12 170.0291 42.23 12 13.1769 9.63
nxy x y
r
nx x ny y
∑−∑ ∑
=
∑−∑ ∑−∑
−
=
−−
CHAPTER 9 │ CORRELATION AND REGRESSION 361
28. (a)
(b)
x y xy x2 y2
1.60 0.78 1.2480 2.5600 0.6084
1.55 0.80 1.2400 2.4025 0.6400
1.44 0.73 1.0512 2.0736 0.5329
1.40 0.72 1.0080 1.9600 0.5184
1.32 0.68 0.8976 1.7424 0.4624
1.23 0.64 0.7872 1.5129 0.4096
()()
() ()
22
22
nxy x y
r
nx x ny y
∑−∑ ∑
=
∑−∑ ∑−∑
29. The correlation coefficient becomes 0.621r≈. The new data entry is an outlier, so the linear
correlation is weaker.
30. The correlation coefficient becomes 0.343r≈− . The new data entry is an outlier, so the linear
correlation is weaker.
362 CHAPTER 9 │ CORREALTION AND REGRESSION
31. 0.623r≈
8 and 0.01
critical value 0.834
0.623 0.834 The correlation is not significant.
or
n
r
α
==
=
≈<⇒
32. 0.955r≈
8 and 0.05
critical value 0.707
0.955 0.707 The correlation is significant.
or
n
r
α
==
=
=>⇒
CHAPTER 9 │ CORRELATION AND REGRESSION 363
33. 0.923r≈
13 and 0.01
critical value 0.684
0.923 0.684 The correlation is significant.
or
n
r
α
==
=
=>⇒
linear correlation exists.
34. 0.831r≈−
12 and 0.05
critical value 0.576
0.831 0.576 The correlation is significant.
or
n
r
α
==
=
=>⇒
364 CHAPTER 9 │ CORREALTION AND REGRESSION
35. 0.828r≈
12 and 0.01
critical value 0.708
0.828 0.708 The correlation is significant.
n
r
α
==
=
≈>⇒
significant linear correlation between earnings per share and dividends per share.
36. 0.981r≈
14 and 0.05
critical value 0.532
0.981 0.532 The correlation is significant.
or
n
r
α
==
=
≈>⇒
37. (a)
CHAPTER 9 │ CORRELATION AND REGRESSION 365
38. (a)
39. The correlation coefficient becomes 0.085r≈. The new rejection regions are 3.499t<− and
3.499t> and the new standardized test statistic is 0.227t≈. So, now you fail to reject 0
H.
40. The correlation coefficient becomes 0.042r≈− . The new rejection regions are 2.447t<− and
2.447t> and the new standardized test statistic is 0.102t≈− . So, you still fail to reject 0
H.
9.2 LINEAR REGRESSION
9.2 Try It Yourself Solutions
1a.
7
88
n
x
=
=
∑
366 CHAPTER 9 │ CORREALTION AND REGRESSION
b.
()()
()
2
2
nxy x y
m
nx x
∑−∑ ∑
=∑−∑
2a. Enter the data.
b. 189.038015
13,497.9583
m
b
≈
≈
9.2 EXERCISE SOLUTIONS
1. A residual is the difference between the observed y-value of a data point and the predicted y–value
on the regression line for the x-coordinate of the data point. A residual is positive when the data
point is above the line, negative when the point is below the line, and zero when the observed y–
value equals the predicted y-value.
2. Positive
3. Substitute a value of x into the equation of a regression line and solve for y.
CHAPTER 9 │ CORRELATION AND REGRESSION 367
17.
x y xy x2
869 60 52,140 755,161
820 50 41,000 672,400
771 50 38,550 594,441
()()
()
()( ) ( )( )
()( ) ( )
2
2
2
9 286,054 6158 405
9 4,350,854 6158
nxy x y
m
nx x
∑−∑ ∑
=∑−∑
−
=−
0.065 0.465ymxb x=+= +
368 CHAPTER 9 │ CORREALTION AND REGRESSION
18.
x y xy x2
1924 174.9 336,507.6 3,701,776
1592 136.9 217,944.8 2,534,464
2413 275.0 663,575.0 5,822,569
()()
()
()( ) ( )( )
()( ) ( )
2
2
2
7 2,233, 453 12,403 1171.6
7 23,260,365 12, 403
nxy x y
m
nx x
∑−∑ ∑
=∑−∑
−
=−
(a)
()
0.123 1450 50.030 128.32y=−=
CHAPTER 9 │ CORRELATION AND REGRESSION 369
19.
x y xy x2
0 40 0 0
1 41 41 1
2 51 102 4
4 48 192 16
()()
()
()( )( )( )
()( )( )
2
2
2
13 4620 60 891
13 346 60
nxy x y
m
nx x
∑−∑ ∑
=∑−∑
−
=−
13 13
⎝⎠ ⎝⎠
7.350 34.617yx=+
370 CHAPTER 9 │ CORREALTION AND REGRESSION
20.
x y xy x2
0 96 0 0
1 85 85 1
2 82 164 4
3 74 222 9
3 95 285 9
()()
()
()( )()( )
()( )()
2
2
2
12 3724 54 908
12 332 54
xxy x y
m
nx x
∑−∑ ∑
=∑−∑
−
=−
CHAPTER 9 │ CORRELATION AND REGRESSION 371
21.
x y xy x2
150 420 63,000 22,500
170 470 79,900 28,900
120 350 42,000 14,400
()
()( )( )( )
()( )( )
2
2
2
10 576,400 1340 4120
10 189, 400 1340
m
nx x
=∑−∑
−
=−
2.472 80.813yx=+
(a)
()
2.472 170 80.813 501.053 milligramsy=+=
372 CHAPTER 9 │ CORREALTION AND REGRESSION
22.
x y xy x2
140 6 840 19,600
200 9 1800 40,000
160 6 960 25,600
170 9 1530 28,900
170 10 1700 28,900
()()
()
()( )( )( )
2
2
2
11 23, 220 1950 127
11 349,900 1950
xxy x y
m
nx x
∑−∑ ∑
=∑−∑
−
=−
(a)
()
0.167 150 18.140 6.91 gramsy=−=
()