CHAPTER 9 │ CORRELATION AND REGRESSION 387
29.
i
x
i
y
i
y
i
yy−
ii
yy− i
yy−
9.4 7.6 7.1252 0.4372 0.4748 0.912
9.2 6.9 7.0736 0.3856 –0.1736 0.212
30. (a) Explained variation
()
21.305
i
yy=∑ − ≈
31. 20.746;r≈ About 74.6% of the variation in the median age of trucks can be explained by the
variation in the median age of cars, and about 25.4% of the variation is unexplained.
32. 0.272;
e
s≈ The standard error of estimate of the median age of trucks for a specific median age
of cars is about 0.272 year.
34. The slope and correlation coefficient will have the same sign because the denominators of both
formulas are always positive while the numerators are always equal.
35a. 20.671r≈
b. 1.780
e
s≈
c. 9.537 19.010y<<
388 CHAPTER 9 │ CORREALTION AND REGRESSION
38. critical value 2.306=±
() ()
22
2
1.328
5.086
374
1.328 14,660 6.771
5.086 10
e
e
m
s
x
m
tx
sn
≈
≈
∑
=∑− ≈ − ≈
Reject
0: 0HM=.
40.
() ()( )
()
()
2
2
2
2
2
23.1
11
10
3.355 138.255 321.847
10 23.1
/67.35 10
ce
x
Etsnxxn
⎛⎞
⎟
⎜⎟
⎜⎟
⎜
⎝⎠
=+ ≈ + ≈
⎡⎤
∑−∑
⎢⎥
⎣⎦ −
()
102.289 321.847 219.558, 424.136bE±⇒ ± =−
CHAPTER 9 │ CORRELATION AND REGRESSION 389
9.4 MULTIPLE REGRESSION
9.4 Try It Yourself Solutions
1a. Enter the data. b.
12
46.385 0.540 4.897yxx=+ −
2ab. (1)
() ()
46.385 0.540 89 4.897 1y=+ −
9.4 EXERCISE SOLUTIONS
1.
12
61,298 57.56 78.45yxx=+ −
(a)
() ()
61,298 57.56 1100 78.45 1090 39,103.5y=+ − = pounds per acre
(b)
() ()
61,298 57.56 1060 78.45 1050 39,939.1y=+ − = pounds per acre
(c)
() ()
61,298 57.56 1300 78.45 1250 38,063.5y=+ − = pounds per acre
(d)
() ()
61,298 57.56 1140 78.45 1120 39,052.4y=+ − = pounds per acre
3.
12
52.2 0.3 4.5yxx=− − +
(a)
() ( )
52.2 0.3 70 4.5 8.6y=− − + =7.5 cubic feet
390 CHAPTER 9 │ CORREALTION AND REGRESSION
4.
123
4016 11.5 7.55 12.5yxxx=− + + +
5.
12
2518.364 126.822 66.360yxx=− + +
(a) 28.489;
e
s= The standard error of estimate of the predicted sales given a specific total
square footage and number of shopping centers is $28.489 billion.
(b)
20.985;r= The multiple regression model explains 98.5% of the variation in y.
7.
12
2518.364 126.822 66.360yxx=− + +
The equation is the same.
8.
12
16.984 0.007 0.524yxx=− − +
10. 2
5, 2, 0.963nkr== =
()
()
2
211
1 0.925
1
adj
rn
rnk
⎡⎤
−−
⎢⎥
=− ≈
⎢⎥
−−
⎢⎥
⎣⎦
CHAPTER 9 │ CORRELATION AND REGRESSION 391
CHAPTER 9 REVIEW EXERCISE SOLUTIONS
1.
2.
3.
4.
392 CHAPTER 9 │ CORREALTION AND REGRESSION
5. 0: 0; : 0
a
HHρρ=≠
0.01, d.f. 2 24nα==−=
02.797t=
6. 0: 0; : 0
a
HHρρ=≠
0.05, d.f. 2 20nα==−=
02.086t=±
linear correlation exists.
7. 0: 0; : 0
a
HHρρ=≠
0.05, d.f. 2 5nα==−=
02.571t=±
()
22
0.912 4.972
110.912
272
r
t
r
n
== ≈
−−
−−
Reject
0
H. There is enough evidence at the 5% level of significance to conclude that a significant
linear correlation exists between passing attempts and passing yards.
8. 0: 0; : 0
a
HHρρ=≠
0.05, d.f. 2 6nα==−=
CHAPTER 9 │ CORRELATION AND REGRESSION 393
9. 0: 0; : 0
a
HHρρ=≠
0.01, d.f. 2 7nα==−=
03.499t=±
()
22
0.338 0.950
11 0.338
292
r
t
r
n
=≈ ≈
−−
−−
Fail to reject 0
H. There is not enough evidence at the 1% level of significance to conclude that a
significant linear correlation exists between brain size and IQ.
11.
0.038 3.529yx=−
0.821r≈ (Strong positive linear correlation)
394 CHAPTER 9 │ CORREALTION AND REGRESSION
14.
0.090 44.675yx=− +
0.984r≈− (Strong negative linear correlation)
16. (a)
()
1.076 4.2 0.299 4.818y=+= hours
(b)
()
1.076 4.5 0.299 5.141y=+= hours
(c)
()
1.076 4.75 0.299 5.41y=+= hours
(d) It is not meaningful to predict the value of y for x = 5 because x = 5 is outside the range of the
original data.
CHAPTER 9 │ CORRELATION AND REGRESSION 395
(c)
()
0.090 289 44.675 18.67y=− + ≈ miles per gallon
(d) It is not meaningful to predict the value of y for x = 407 because x = 407 is outside the range
of the original data.
21.
()
2
20.642 0.412r=≈
About 41.2% of the variation in y is explained.
About 58.8% of the variation in y is unexplained.
22.
()
2
20.795 0.632r=≈
About 63.2% of the variation in y is explained.
About 36.8% of the variation in y is unexplained.
24. (a) 20.446r≈
About 44.6% of the variation in y is explained.
About 55.4% of the variation in y is unexplained.
(b) 235.079
e
s≈
The standard error of the price for a specific area is about $235.08.
396 CHAPTER 9 │ CORREALTION AND REGRESSION
27.
()
0.086 45 10.450 6.580y=− + =
()
()
()()() ()
()()
22
22
2
7 45 337 / 7
11
1 2.571 0.622 1 77 18,563 337
1.7127
ce
nx x
Ets nnx x
−−
=++ ≈ ++
∑−∑ −
≈
28.
()
0.090 265 44.675 20.825y=− + =
()
()
()()() ()
()()
22
2 2
2
7 265 1516 / 7
11
1 2.571 1.476 1 77 369,382 1516
ce
nx x
Ets nnx x
−−
=++ ≈ ++
∑−∑ −
CHAPTER 9 │ CORRELATION AND REGRESSION 397
29.
()
0.414 39.9 37.147 20.628y=− + ≈
30.
()
1.454 900 532.053 776.547y=−=
()
()
()()( ) ()
()()
22
2 2
2
18 900 13,203/18
11
1 2.921 235.079 1 18 18 10,021,083 13, 203
732.481
ce
nx x
Ets nnx x
−−
=++ ≈ ++
∑−∑ −
≈
776.547 732.481 776.547 732.481
44.066 1509.028
yE y yE
y
y
−<<+
−<< +
<<
CHAPTER 9 QUIZ SOLUTIONS
1.
398 CHAPTER 9 │ CORREALTION AND REGRESSION
2. 0.993;r≈Strong positive linear correlation; public school classroom teachers’ salaries increase
as public school principals’ salaries increase.
Reject 0
H. There is enough evidence at the 5% level of significance to conclude that a significant
correlation exists.
4.
x y xy x2
62.5 37.3 2331.25 3906.25
71.9 41.4 2976.66 5169.61
74.4 42.2 3139.68 5535.36
77.8 43.7 3399.86 6052.84
()()
()
()()()
()()
22
2
11 39,439.08 871.9 493.7 3372.74 0.490827
6871.54
11 69,734.65 871.9
nxy x y
m
nx x
∑−∑ ∑ −
== =≈
∑−∑ −
()
493.7 871.9
0.490827 5.9771
11 11
yx
bymx m
nn
⎛⎞ ⎛ ⎞
∑∑
⎟⎟
⎜⎜
=− = − ≈ − ≈
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎝⎠ ⎝ ⎠
0.491 5.977ymxb x=+= +
CHAPTER 9 │ CORRELATION AND REGRESSION 399
6.
()
2
20.993 0.986r≈≈
About 98.6% of the variation in the average annual salaries of public school classroom teachers
can be explained by the average annual salaries of public school principals, and about 1.4% of the
variation is unexplained.
8.
()
0.491 85.75 5.977 48.080y=+≈
()
()
()()() ()
()()
22
2 2
2
11 85.75 871.9 /11
11
1 2.262 0.490 1 11 11 69,734.65 871.9
1.193
ce
nx x
Ets nnx x
−−
=++ ≈ ++
∑−∑ −
≈
yE y yE
−<<+