Coefficient of Determination Worksheet Answers
1. An economist is studying the job market in Denver – area neighborhoods. Let 𝑥
represent the total number of jobs in a given neighborhood, and let 𝑦 represent the
number of entry – level jobs in the same neighborhood. A sample of six Denver –
area neighborhoods gave the following information ( units in hundreds of jobs )
𝑥
16
33
50
28
50
25
𝑦
2
3
6
5
9
3
a) Draw a scatter diagram
10
8
6
4
(33.67,4.67)
2
(20,2.5)
10
30
20
40
50
b) Complete the table
𝑥
𝑦
𝑥2
𝑦2
𝑥𝑦
16
2
256
4
32
33
3
1089
9
99
50
6
2500
36
300
28
5
784
25
140
50
9
2500
81
450
25
∑𝑥 = 202
3
625
∑𝑥 2 = 7754
9
∑𝑦 2 = 164
75
∑𝑥𝑦 = 1096
∑𝑦 = 28
c) Compute 𝑟
𝑟=
𝑟=
𝑛 ∑𝑥𝑦−(∑𝑥)(∑𝑦)
√𝑛 ∑𝑥 2 −(∑𝑥)2
√𝑛 ∑𝑦 2 −(∑𝑦)2
6576−5656
√46452−40804 √984−784
=
(6)(1096)−(202)(28)
=
√(6)(7754)−(202)2 √(6)(164)−(28)2
920
920
√5648 √200
= (75.15)(14.14) =
920
1062.62
= 0.866
d) Find the least – squares equation
𝑥̅ =
𝑏=
∑𝑥
𝑛
=
202
6
= 33.67
𝑛 ∑𝑥𝑦−(∑𝑥)(∑𝑦)
𝑛
∑𝑥 2 −(∑𝑥)2
=
𝑦̅ =
(6)(1096)−(202)(28)
(6)(7754)−40804
∑𝑦
=
𝑛
=
920
5648
28
6
= 4.67
= 0.16
𝑎 = 𝑦̅ − 𝑏𝑥̅ = 4.67 − (0.16)(33.67) = 4.67 − 5.39 = −0.72
𝑦̂ = 𝑎 + 𝑏𝑥
𝑦̂ = −0.72 + 0.16𝑥
e) Graph the least squares equation on your scatter diagram. Be sure to use the
point (𝑥̅ , 𝑦̅) as one of the points on the line
** see above
𝑦̂(20) = −0.72 + 0.16(20) = −0.72 + 3.2 = 2.5
f) Find the coefficient of determination 𝑟 2 . What percentage of variation in 𝑦 can be
explained by the corresponding variation in 𝑥 and the least – squares line ?
𝑟 = 0.866
𝑟 2 = 0.8662 = 0.75
75% of variation can be explained
g) For a neighborhood with 𝑥 = 40 jobs, how many predicted to be entry – level ?
𝑝̂ = −0.72 + 0.16𝑥
𝑝̂ (40) = −0.72 + 0.16(40) = −0.72 + 6.4 = 5.7
2. Do heavier cars really use more gasoline ? Suppose a car is chosen at random. Let 𝑥
be the weight of the car ( in hundreds of pounds ) and let 𝑦 be the miles per gallon.
The following information is based on data taken from Consumer Reports.
𝑥
𝑦
a)
27
30
44
19
32
24
47
13
23
29
40
17
34
21
52
14
Draw a scatter diagram
30
25
(37.38,20.88)
20
15
(50,13.3)
10
10
20
30
40
50
b) Complete the table
𝑥
𝑦
𝑥2
𝑦2
𝑥𝑦
27
30
729
900
810
44
19
1936
361
836
32
24
1024
576
768
47
13
2209
169
611
23
29
529
841
667
40
17
1600
289
680
34
21
1156
441
714
52
∑𝑥 = 299
14
2704
∑𝑦 = 167
∑𝑥 2 = 11887
196
728
∑𝑦 2 = 3773 ∑𝑥𝑦 = 5814
c) Compute 𝑟
𝑟=
𝑟=
𝑛 ∑𝑥𝑦 − (∑𝑥)(∑𝑦)
√𝑛 ∑𝑥 2 − (∑𝑥)2 √𝑛 ∑𝑦 2 − (∑𝑦)2
46512−49933
√95096−89401 √30184−27889
=
=
(8)(5814) − (299)(167)
√(8)(11887) − (299)2 √(8)(3773) − (167)2
−3421
−3421
= (75.47)(47.91) =
√5695 √2295
−3421
3615.77
= −0.95
d) Find the least – squares equation
𝑥̅ =
𝑏=
∑𝑥
𝑛
=
299
8
= 37.38
𝑛 ∑𝑥𝑦−(∑𝑥)(∑𝑦)
𝑛
∑𝑥 2 −(∑𝑥)2
=
𝑦̅ =
(8)(5814)−(299)(167)
(8)(11887)−89401
∑𝑦
𝑛
=
=
167
8
−3421
5695
= 20.88
= −0.60
𝑎 = 𝑦̅ − 𝑏𝑥̅ = 20.88 − (−0.60)(37.38) = 20.88 + 22.43 = 43.31
𝑦̂ = 𝑎 + 𝑏𝑥
𝑦̂ = 43.31 − 0.60𝑥
e) Graph the least squares equation on your scatter diagram. Be sure to use the
point (𝑥̅ , 𝑦̅) as one of the points on the line
** see above graph
𝑦̂(50) = 43.31 − 0.6(50) = 13.31
f) Find the coefficient of determination 𝑟 2 . What percentage of variation in 𝑦 can be
explained by the corresponding variation in 𝑥 and the least – squares line ?
𝑟 = −0.95
𝑟 2 = (−0.95)2 = 0.9025
90.2 % of the variation can be explained
g) Suppose a car weighs 𝑥 = 38 ( hundred pounds ). What does the least – squares
line forecast for 𝑦 = miles per gallon?
𝑦̂ = 𝑎 + 𝑏𝑥
𝑦̂(38) = 44.31 − 0.60(38) = 44.31 − 22.8 = 21.51 mpg
3. It is thought that basketball teams that make too many fouls in a game tend to lose
the game even if they otherwise play well. Let 𝑥 be the number of fouls that were
more than the number of fouls made by the other team. Let 𝑦 be the percentage of
times the team with the larger number of fouls won the game.
a)
𝑥
0
2
5
6
𝑦
50
45
33
26
Draw a scatter diagram
(1,47.3)
50
45
(3.25,38.5)
40
35
30
25
1
2
4
3
5
6
b) Complete the table
𝑥
𝑦
𝑥2
𝑦2
𝑥𝑦
0
50
0
2500
0
2
45
4
2025
90
5
33
25
1089
165
6
∑𝑥 = 13
26
36
∑𝑥 2 = 65
676
156
∑𝑥𝑦 = 411
∑𝑦 = 154
∑𝑦 2 = 6290
c) Compute 𝑟
𝑟=
𝑟=
𝑛 ∑𝑥𝑦 − (∑𝑥)(∑𝑦)
√𝑛 ∑𝑥 2 − (∑𝑥)2 √𝑛 ∑𝑦 2 − (∑𝑦)2
1644 − 2002
√260 − 169 √25160 − 23716
=
=
(4)(411) − (13)(154)
√(4)(65) − (13)2 √(4)(6290) − (154)2
−358
√91 √1444
=
−358
−358
=
= −0.98
(9.54)(38) 362.52
d) Find the least – squares equation
𝑥̅ =
𝑏=
∑𝑥
𝑛
=
13
4
= 3.25
𝑛 ∑𝑥𝑦−(∑𝑥)(∑𝑦)
𝑛
∑𝑥 2 −(∑𝑥)2
=
𝑦̅ =
∑𝑦
𝑛
=
154
(4)(411)−(13)(154)
(4)(65)−169
4
=
= 38.5
−358
91
= −3.93
𝑎 = 𝑦̅ − 𝑏𝑥̅ = 38.5 − (−3.93)(3.25) = 38.5 + 12.77 = 51.27
e)
𝑦̂ = 𝑎 + 𝑏𝑥
𝑦̂ = 51.27 − 3.93𝑥
f) Graph the least squares equation on your scatter diagram. Be sure to use the
point (𝑥̅ , 𝑦̅) as one of the points on the line
** see graph above 𝑦̂(1) = 51.27 − 3.93(1) = 51.27 − 3.93 = 47.3
g) Find the coefficient of determination 𝑟 2 . What percentage of variation in 𝑦 can be
explained by the corresponding variation in 𝑥 and the least – squares line ?
𝑟 = −0.98
𝑟 2 = (−0.98)2 = 0.96
96% of the variation can be explained
h) If a team had 𝑥 = 4 over and above the opposing team, what does the least –
squares equation forecast for 𝑦?
𝑦̂ = 51.27 − 3.93𝑥
𝑦̂(4) = 51.27 − 3.93(4) = 51.27 − 15.72 = 35.55 % of the time won the game
4. Let 𝑥 be per capita income in thousands of dollars. Let 𝑦 be the number of medical
doctors per 10,000 residents. Six small cities in Oregon gave the following
information about 𝑥 and 𝑦.
𝑥
8.6
9.3
10.1
8.0
8.3
8.7
𝑦
9.6
18.5
20.9
10.2
11.4
13.1
a) Draw a scatter diagram
21
(9.5,17.81)
18
15
(8.83,13.95)
12
9
8
8.5
9
9.5
10
b) Complete the table
𝑥
𝑦
𝑥2
𝑦2
𝑥𝑦
8.6
9.6
73.96
92.16
82.56
9.3
18.5
86.49
342.25
172.05
10.1
20.9
102.01
436.81
211.09
8.0
10.2
64.0
104.04
81.6
8.3
11.4
68.89
129.96
94.62
8.7
13.1
75.69
171.61
113.97
∑𝑥 = 53
∑𝑦 = 83.7
∑𝑥 2 = 471.04
∑𝑦 2 = 1276.83
∑𝑥𝑦 = 755.89
c) Compute 𝑟
𝑟=
𝑟=
𝑛 ∑𝑥𝑦 − (∑𝑥)(∑𝑦)
√𝑛 ∑𝑥 2 − (∑𝑥)2 √𝑛 ∑𝑦 2 − (∑𝑦)2
(6)(755.89) − (53)(83.7)
=
4535.34 − 4436.1
√2826.24 − 2809 √7660.98 − 7005.69
√(6)(471.04) − (53)2 √(6)(1276.83) − (83.7)2
=
99.24
√17.24 √655.29
=
99.24
99.24
=
= 0.93
(4.15)(25.6) 106.24
d) Find the least – squares equation
𝑥̅ =
𝑏=
∑𝑥
𝑛
=
53
6
= 8.83
𝑛 ∑𝑥𝑦−(∑𝑥)(∑𝑦)
𝑛 ∑𝑥 2 −(∑𝑥)2
=
𝑦̅ =
∑𝑦
𝑛
=
83.7
6
(6)(755.89)−(53)(83.7)
(6)(471.04)−2809
= 13.95
=
99.24
17.24
= 5.76
𝑎 = 𝑦̅ − 𝑏𝑥̅ = 13.95 − (5.76)(8.83) = 13.95 − 50.86 = −36.91
a)
𝑦̂ = 𝑎 + 𝑏𝑥
𝑦̂ = −36.91 + 5.76𝑥
e) Graph the least squares equation on your scatter diagram. Be sure to use the
point (𝑥̅ , 𝑦̅) as one of the points on the line
** see graph above
𝑦̂(9.5) = −36.91 + 5.76(9.5) = 17.81
f) Find the coefficient of determination 𝑟 2 . What percentage of variation in 𝑦 can be
explained by the corresponding variation in 𝑥 and the least – squares line ?
𝑟 = 0.93
𝑟 2 = (0.93)2 = 0.87
87% of the variation can be explained
g) Suppose a small city in Oregon has a per capita income of 10 thousand dollars.
What is the predicted number of doctors per 10,000 residents?
𝑦̂ = −36.91 + 5.76𝑥
𝑦̂(10) = −36.91 + 5.76(10) = −36.91 + 57.6 = 20.7