CHAPTER 3 Calculating Descriptive Statistics 3 Calculating Descriptive Statistics 3.1 a) 1 18 12 7...

3-1 Copyright ©2015 Pearson Education, Inc.

CHAPTER 3 Calculating Descriptive Statistics

3.1

a) 1 18 12 7 10 15 12 6 12 14 10611.8

9 9

n

ii

xx

n= + + + + + + + += = = =

b)

( ) ( )0.5 0.5 9 4.5i n= = =

6 7 10 12 12 12 14 15 18 Median = 12

c) Mode = 12 d) Because the mean and median are very close, this distribution’s shape would be described as symmetrical. 3.2

a) 1 6 14 17 1 16 17 6 19 9612.0

8 8

n

ii

xx

n= + + + + + + += = = =

b)

( ) ( )0.5 0.5 8 4i n= = =

1 6 6 14 16 17 17 19

Median = ( )14 16 / 2 15+ =

c) Mode = 6, 17 d) Because the mean is lower than the median, this distribution’s shape would be described as left-skewed. 3.3

a) 1 14 62 0 20 26 38 8 16824

7 7

n

ii

xx

n= + + + + + += = = =

3-2 Chapter 3

Copyright ©2015 Pearson Education, Inc.

b)

( ) ( )0.5 0.5 7 3.5i n= = =

0 8 14 20 26 38 62 Median = 20

c) There is no mode. d) Because the mean is higher than the median, this distribution’s shape would be described as right-skewed. 3.4

a) ( ) ( )1

25 18 7 10 34 12 6811.3

6 6

n

ii

xx

n= + + − + + + −

= = = =

b)

( ) ( )0.5 0.5 6 3i n= = =

-12 -7 10 18 25 34

Median = ( )10 18 / 2 14+ =

c) There is no mode. d) Because the mean is lower than the median, this distribution’s shape would be described as left-skewed. 3.5

a) 1 9.1 9.0 8.9 9.0 ... 7.8 7.9 211.98.48

25 25

n

ii

xx

n= + + + + + += = = =

b)

( ) ( )0.5 0.5 25 12.5i n= = =

7.8 7.8 7.8 7.9 7.9 8.1 8.1 8.2 8.2 8.2 8.2 8.3 8.3 8.5 8.6 8.9 8.9 9.0 9.0 9.0 9.0 9.0 9.0 9.1 9.1 Median = 8.3

c) Mode = 9.0 d) Because the mean is higher than the median, this distribution’s shape would be described as right-skewed. 3.6

a) 1 $7.97 $9.57 $8.11 ... $11.61 $11.32 $107.45$8.95

12 12

n

ii

xx

n= + + + + += = = =

Calculating Descriptive Statistics 3-3


b)

( ) ( )0.5 0.5 12 6i n= = =

$7.34 $7.35 $7.97 $7.99 $8.11 $8.18 $8.83 $9.32 $9.57 $9.86 $11.32 $11.61

Median = ( )$8.18 $8.83 / 2 $8.505+ =

c) There is no mode. d) Because the mean is higher than the median, this distribution’s shape would be described as right-skewed. 3.7

( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )14 0 8 1 8 2 12 3 5 4 3 5 95

1.914 8 8 12 5 3 50

i i

i

w xx

w+ + + + +

= = = =+ + + + +

3.8

a) 1 17.7 4.0 7.0 ... 1.3 8.6 189.413.53

14 14

n

ii

xx

n= + + + + += = = =

b)

( ) ( )0.5 0.5 14 7i n= = =

1.3 1.4 2.1 4.0 5.8 6.3 7.0 8.4 8.6 11.2 16.2 17.7 19.6 79.8

Median = ( )7.0 8.4 / 2 7.7+ =

c) There is no mode. d) Because 79.8 could be considered an outlier, the median would be the measurement that best describes the central tendency of these data. e) Because the mean is higher than the median, this distribution’s shape would be described as right-skewed. 3.9

a) 1 $1.23 $1.24 $1.25 ... $1.44 $1.44 $32.11$1.34

24 24

n

ii

xx

n= + + + + += = = =

b)

( ) ( )0.5 0.5 24 12i n= = =

1.23 1.24 1.25 1.28 1.28 1.29 1.29 1.30 1.31 1.32 1.32 1.32 1.32 1.33 1.35 1.36 1.37 1.38 1.40 1.43 1.43 1.43 1.44 1.44

Median = ( )1.32 1.32 / 2 1.32+ =

3-4 Chapter 3


c) Mode = 1.32 d) Because there does not appear to be any an outliers, the mean or the median would adequately

describe the central tendency of these data. e) Because the mean is higher than the median, this distribution’s shape would be described as right-skewed. 3.10 a)

1986-1993: 1 3 49 32 33 39 22 42 9 22928.6

8 8

n

ii

xx

n= + + + + + + += = = =

1994-2001: 1 9 39 52 58 70 65 32 29 35444.3

8 8

n

ii

xx

n= + + + + + + += = = =

b)

( ) ( )0.5 0.5 8 4i n= = =

1986-1993: 3 9 22 32 33 39 42 49

Median = ( )32 33 / 2 32.5+ =

1994-2001: 9 29 32 39 52 58 65 70

Median = ( )39 52 / 2 45.5+ =

c) There is no mode in either data set. d) Because the mean is lower than the median for the first 8 years, this distribution’s shape would be described as left-skewed. Because the mean is close to the median for the last 8 years, this distribution’s shape would be described as symmetrical. e) The mean for the second 8 years is much higher than the mean of the first 8-year period. There seems to be enough evidence to doubt his claim. 3.11 a) Range 18 6 12= − =



b)

ix 2ix

18 324 12 144 7 49 10 100 15 225 12 144 6 36 12 144 14 196

9

1

106ii

x=

= 9

2

1

1,362ii

x=

=

( )

2

22 1

2 1

1061,362 1,362 1, 248.49 14.2

1 9 1 8

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

c) 14.2 3.77s = =

3.12 a) Range 23 5 18= − = b)

ix 2ix

5 25 11 121 17 289 23 529 14 196 16 256 11 121 17 289

8

1

114ii

x=

= 8

2

1

1,826ii

x=

=

( )

2

22 1

2 1

1141,826 1,826 1,624.58 28.8

1 8 1 7

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

c) 28.8 5.37s = =

3-6 Chapter 3


3.13 a) Range 41 0 41= − = b)

ix 2ix

14 196 41 1,681 0 0 20 400 26 676 38 1,444 8 64

7

1

147ii

x=

= 7

2

1

4, 461ii

x=

=

( )

2

22 1

2 1

1474, 461 4, 461 3,0877 196.3

7 7

n

ini

ii

xx

NN

σ

=

=

− − −= = = =

c) 196.3 14.01σ = = 3.14

a) ( )Range 34 12 46= − − = b)

ix 2ix

25 625 18 324 -7 49 10 100 34 1,156 -12 144

6

1

68ii

x=

= 6

2

1

2,398ii

x=

=

( )

2

22 1

2 1

682,398 2,398 770.76 271.2

6 6

n

ini

ii

xx

NN

σ

=

=

− − −= = = =

c) 271.2 16.47σ = =



3.15 a) Range 10 5 5= − = b)

ix 2ix

8 64 8 64 7 49 10 100 6 36 9 81 5 25 7 49 6 36 8 64

10

1

74ii

x=

= 10

2

1

568ii

x=

=

( )

2

22 1

2 1

74568 568 547.610 2.3

1 10 1 9

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

c) 2.3 1.52s = =

3.16 a) Range 6 0 6= − = b)

ix 2ix

2 4 4 16 0 0 1 1 6 36 1 1 3 9 2 4 3 9

9

1

22ii

x=

= 9

2

1

80ii

x=

=

3-8 Chapter 3


( )

2

22 1

2 1

2280 80 53.89 3.3

1 9 1 8

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

c) 3.3 1.82s = =

3.17 a) Range 6 1 5= − = b)

ix 2ix

5 25 1 1 2 4 6 36 4 16 4 16 3 9 5 25

8

1

30ii

x=

= 8

2

1

132ii

x=

=

( )

2

22 1

2 1

30132 132 112.58 2.8

1 8 1 7

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

c) 2.8 1.67s = =

3.18 a) Range 18 3 15= − =



b)

ix 2ix

14 196 5 25 6 36 11 121 18 324 10 100 10 100 12 144 3 9 15 225

10

1

104ii

x=

= 10

2

1

1, 280ii

x=

=

( )

2

22 1

2 1

1041, 280 1, 280 1,081.610 22.0

1 10 1 9

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

c) 22.0 4.69s = =

3.19 a) Range 44 32 12= − = b)

ix 2ix

36 1,296 32 1,024 40 1,600 35 1,225 32 1,024 41 1,681 35 1,225 44 1,936 38 1,444 37 1,369 40 1,600

11

1

410ii

x=

= 11

2

1

15,424ii

x=

=

3-10 Chapter 3


( )

2

22 1

2 1

41015, 424 15, 424 15, 281.811 14.22

1 11 1 10

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

c) 14.22 3.77s = =

3.20 a) Set 1 Set 2

ix 2ix ix 2

ix

4 16 2 4 7 49 11 121 6 36 14 196 9 81 4 16 5 25 3 9

5

1

31ii

x=

= 5

2

1

207ii

x=

=

5

1

34ii

x=

= 5

2

1

346ii

x=

=

Set 1: 1 31

6.25

n

ii

xx

n== = =

Set 2:

1 346.8

5

n

ii

xx

n== = =

Set 1:

( )

2

22 1

1

31207 207 192.25 1.92

1 5 1 4

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

Set 2:

( )234

346 346 231.25 5.365 1 4

s− −= = =−

Set 1: ( ) ( )1.92100 100 31.0%

6.2

sCVx

= = =

Set 2: ( )5.36100 78.8%

6.8CV = =

b) Set 2 has more variability because it has a higher coefficient of variation.



3.21 a) Set 1 Set 2

ix 2ix ix 2

ix

11 121 3 9 21 441 8 64 15 225 0 0 24 576 5 25 20 400 6 36

5

1

91ii

x=

= 5

2

1

1,763ii

x=

=

5

1

22ii

x=

= 5

2

1

134ii

x=

=

Set 1: 1 91

18.25

n

ii

xx

n== = =

Set 2:

1 224.4

5

n

ii

xx

n== = =

Set 1:

( )

2

22 1

1

911,763 1,763 1,656.25 5.17

1 5 1 4

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

Set 2:

( )222

134 134 96.85 3.055 1 4

s− −= = =

−

Set 1: ( ) ( )5.17100 100 28.4%

18.2

sCVx

= = =

Set 2: ( )3.05100 69.3%

4.4CV = =

b) Set 2 has more variability because it has a higher coefficient of variation. 3.22

a) 50 40

0.8312

x xzs− −= = =

b) 65 40

2.0812

x xzs− −= = =

c) 35 40

0.4212

x xzs− −= = = −

d) 10 40

2.5012

x xzs− −= = = −

3-12 Chapter 3


3.23

ix 2ix

9 81 7 49 15 225 10 100 4 16 12 144

6

1

57ii

x=

= 6

2

1

615ii

x=

=

1 579.5

6

n

ii

xx

n== = =

( )

2

22 1

1

57615 615 541.56 3.83

1 6 1 5

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

a) 13 9.5

0.913.83

x xzs− −= = =

b) 16 9.5

1.703.83

x xzs− −= = =

c) 8 9.5

0.393.83

x xzs− −= = = −

d) 4 9.5

1.443.83

x xzs− −= = = −

3.24

a) $699 $800

1.35$75

x xzs− −= = = −

b) $949 $800

1.99$75

x xzs− −= = =

c) $625 $800

2.33$75

x xzs− −= = = −

d) $849 $800

0.65$75

x xzs− −= = =

e) $999 $800

2.65$75

x xzs− −= = =



3.25

ix 2ix

1 1 4 16 6 36 0 0 4 16 3 9 10 100 0 0 4 16

9

1

32ii

x=

= 9

2

1

194ii

x=

=

1 323.6

9

n

ii

xx

n== = =

( )

2

22 1

1

32194 194 113.89 3.17

1 9 1 8

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

a) 1 3.6

0.823.17

x xzs− −= = = −

b) 3 3.6

0.193.17

x xzs− −= = = −

c) 7 3.6

1.073.17

x xzs− −= = =

d) 12 3.6

2.653.17

x xzs− −= = =

3.26

a) ( ) ( )$40,000100 100 12.3%

$325,000CV σ

μ= = =

b) $310,000 $325,000

0.38$40,000

xz μσ− −= = = −

3-14 Chapter 3


c)

( ) ( )( ) ( )

$325,000 2 $40,000 $405,000

$325,000 2 $40,000 $245,000

x zx

x

μ σ= += + =

= − = d)

( ) ( )( ) ( )

$325,000 4 $40,000 $485,000

$325,000 4 $40,000 $165,000

x zx

x

μ σ= += + =

= − =

3.27 a)

ix 2ix ix 2

ix

0 0 20 400 18 324 32 1,024 19 361 27 729 33 1,089 32 1,024 38 1,444 30 900 39 1,521 20 400 35 1,225 18 324 31 961 28 784 22 484 22 484

18

1

464ii

x=

= 18

2

1

13, 478ii

x=

=

1 46425.8

18

n

ii

xx

n== = =

( )

2

22 1

1

46413, 478 13, 478 11,960.918 9.45

1 18 1 17

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

Two standard deviations:

( )( )( )( )

25.8 2 9.45 44.7

25.8 2 9.45 6.9

x x zsx

x

= += + =

= − = 17 out of 18 observations fall within these limits (17/18 = 94.4%). Chebyshev’s Theorem claims that at least 75% of the observations will fall within two standard deviation of the mean.



Three standard deviations:

( )( )( )( )

25.8 3 9.45 54.2

25.8 3 9.45 2.6

x x zsx

x

= += + =

= − = − 18 out of 18 observations fall within these limits (18/18 = 100%). Chebyshev’s Theorem claims that at least 89% of the observations will fall within three standard deviation of the mean. 3.28

a) ( ) ( )2.8100 100 8.9%

31.6CV σ

μ= = =

b) 28 31.6

1.292.8

xz μσ− −= = = −

c)

( ) ( )( ) ( )

31.6 3 2.8 40.0

31.6 3 2.8 23.2

x zx

x

μ σ= += + =

= − = d)

( )( )( )( )

31.6 4 2.8 42.8

31.6 4 2.8 20.4

x zx

x

μ σ= += + =

= − =

e)

2

2

2

11 0.80

10.2

5

2.24

z

zzz

− =

=

==

( )( )( )( )

31.6 2.24 2.8 37.9

31.6 2.24 2.8 25.3

x zx

x

μ σ= += + =

= − =

3-16 Chapter 3


3.29 Values Midpoint 0 to under 2 1 2 to under 4 3 4 to under 6 5 6 to under 8 7

( )

( )( ) ( )( ) ( )( ) ( )( )4 1 2 3 9 5 7 7 1044.7

4 2 9 7 22

i if mx

n

x

≈

+ + +≈ = =

+ + +

Value mi fi x ( )−im x ( )−2

im x ( )−2

i im x f

0 to under 2 1 4 4.7 -3.7 13.69 54.76 2 to under 4 3 2 4.7 -1.7 2.89 5.78 4 to under 6 5 9 4.7 0.3 0.09 0.81 6 to under 8 7 7 4.7 2.3 5.29 37.03

22in f= = ( )298.38i im x f− =

( )2

1 98.382.16

1 22 1

k

i ii

m x fs

n=

−≈ = =

− −

3.30 Values Midpoint 5 to under 10 7.5 10 to under 15 12.5 15 to under 20 17.5 20 to under 25 22.5 25 to under 30 27.5

( )

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )15 7.5 9 12.5 21 17.5 24 22.5 6 27.5 1, 297.517.3

15 9 21 24 6 75

i if mx

n

x

≈

+ + + +≈ = =

+ + + +




im x ( )−2

i im x f

5 to under 10 7.5 15 17.3 -9.8 96.04 1,440.6 10 to under 15 12.5 9 17.3 -4.8 23.04 207.36 15 to under 20 17.5 21 17.3 0.2 0.04 0.84 20 to under 25 22.5 24 17.3 5.2 27.04 648.96 25 to under 30 27.5 6 17.3 10.2 104.04 624.24

75in f= = ( )22,922i im x f− =

( )2

1 2,9226.28

1 75 1

k

i ii

m x fs

n=

−≈ = =

− −

3.31 Values Midpoint 10 to under 30 20 30 to under 50 40 50 to under 70 60 70 to under 90 80

( )

( )( ) ( )( ) ( )( ) ( )( )26 20 35 40 51 60 33 80 7,62052.6

26 35 51 33 145

i if mx

n

x

≈

+ + +≈ = =

+ + +


im x ( )−2

i im x f

10 to under 30 20 26 52.6 -32.6 1,062.76 27,631.76 30 to under 50 40 35 52.6 -12.6 158.76 5,556.6 50 to under 70 60 51 52.6 7.4 54.76 2,792.76 70 to under 90 80 33 52.6 27.4 750.76 24,775.08

145in f= = ( )260,756.2i im x f− =

( )2

1 60,756.220.54

1 145 1

k

i ii

m x fs

n=

−≈ = =

− −

3-18 Chapter 3


3.32 Income Midpoint $20 to under $30 $25 $30 to under $40 $35 $40 to under $50 $45 $50 to under $60 $55 $60 to under $70 $65

( )

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )67 $25 111 $35 125 $45 21 $55 38 $65 14,810$40.9

67 111 125 21 38 362

i if mx

n

x

≈

+ + + +≈ = =

+ + + +


im x ( )−2

i im x f

$20 to under $30 $25 67 40.9 -15.9 252.81 16,938.27 $30 to under $40 $35 111 40.9 -5.9 34.81 3,863.91 $40 to under $50 $45 125 40.9 4.1 16.81 2,101.25 $50 to under $60 $55 21 40.9 14.1 198.81 4,175.01 $60 to under $70 $65 38 40.9 24.1 580.81 22,070.78

362in f= = ( )249,149.22i im x f− =

( )2

2 1 49,149.22136.15

1 362 1

136.15 $11.67

n

i ii

m x fs

ns

=

−≈ = =

− −≈ =

3.33 Patients Midpoint 20 to under 40 30 40 to under 60 50 60 to under 80 70 80 to under 100 90 100 to under 120 110



( )

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )10 30 16 50 25 70 65 90 34 110 12, 44082.9

10 16 25 65 34 150

i if mx

n

x

≈

+ + + +≈ = =

+ + + +


im x ( )−2

i im x f

20 to under 40 30 10 82.9 -52.9 2,798.41 27,984.10 40 to under 60 50 16 82.9 -32.9 1,082.41 17,318.56 60 to under 80 70 25 82.9 -12.9 166.41 4,160.25 80 to under 100 90 65 82.9 7.1 50.41 3,276.65 100 to under 120 110 34 82.9 27.1 734.41 24,969.94

150in f= = ( )277,709.5i im x f− =

( )2

2 1 77,709.5521.54

1 150 1

521.54 22.84

k

i ii

m x fs

ns

=

−≈ = =

− −≈ =

3.34 Patients Midpoint 0 to under 10 5 10 to under 20 15 20 to under 30 25 30 to under 40 35 40 to under 50 45 50 to under 60 55

( )

( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )6 5 22 15 10 25 18 35 28 45 16 55 3,38033.8

6 22 10 18 28 16 100

i if mx

n

x

≈

+ + + + +≈ = =

+ + + + +

3-20 Chapter 3



im x ( )−2

i im x f

0 to under 10 5 6 33.8 -28.8 829.44 4,976.64 10 to under 20 15 22 33.8 -18.8 353.44 7,775.68 20 to under 30 25 10 33.8 -8.8 77.44 774.40 30 to under 40 35 18 33.8 1.2 1.44 25.92 40 to under 50 45 28 33.8 11.2 125.44 3,512.32 50 to under 60 55 16 33.8 21.2 449.44 7,191.04

100in f= = ( )224,256i im x f− =

( )2

2 1 24,256245.01

1 100 1

245.01 15.65

k

i ii

m x fs

ns

=

−≈ = =

− −≈ =

3.35 a)

ix 2ix ix 2

ix

13 169 44 1,936 27 729 39 1,521 26 676 29 841 44 1,936 44 1,936 30 900 38 1,444 39 1,521 47 2,209 40 1,600 34 1,156 34 1,156 40 1,600 45 2,025 20 400 44 1,936 12 144

24 576 10 100 32 1,024

23

1

755ii

x=

= 23

2

1

27,535ii

x=

=

1 75532.8

23

N

ii

x

Nμ == = =



( )

2

22 1

1

75527,535 27,535 24,783.723 10.94

23 23

N

iki

ii

xx

NN

σ

=

=

− − −= = = =

b) Home Runs Frequency Midpoint 10 to under 20 3 15 20 to under 30 5 25 30 to under 40 7 35 40 to under 50 8 45 c)

( )

( )( ) ( )( ) ( ) ( ) ( )( )3 15 5 25 7 35 8 45 77533.7

3 5 7 8 23

i if mN

μ

μ

≈

+ + +≈ = =

+ + +

Value mi fi μ ( )μ−im ( )μ− 2

im ( )μ− 2

i im f

10 to under 20 15 3 33.7 -18.7 349.69 1,049.07 20 to under 30 25 5 33.7 -8.7 75.69 378.45 30 to under 40 35 7 33.7 1.3 1.69 11.83 40 to under 50 45 8 33.7 11.3 127.69 1,021.52

23iN f= = ( )22,460.87i im fμ− =

( )2

1 2,460.8710.34

23

k

i ii

m f

N

μσ =

−≈ = =

d) The mean and standard deviations of the grouped data assume that all the values within each group fall at the midpoint of the class, which we can observe is not true by examining each of the data points. 3.36 17 20 30 38 54 55 66 76

a) ( ) ( )208 1.6

100 100

Pi n= = =

20th percentile = 20

3-22 Chapter 3


b) ( ) ( )408 3.2

100 100

Pi n= = =


c) ( ) ( )608 4.8

100 100

Pi n= = =


3.37 5 8 13 18 20 22 30 33 39

a) ( ) ( )159 1.35

100 100

Pi n= = =

15th percentile = 8

b) ( ) ( )359 3.15

100 100

Pi n= = =


c) ( ) ( )859 7.65

100 100

Pi n= = =


3.38

5 10 13 15 37 40 43 45 67 71

a)

( ) ( )

( ) ( )

Number of values below X 0.5Percentile Rank = 100

Total number of values8 0.5

Percentile Rank 50 = 100 8510

+

+ =

b) ( ) ( )4 0.5Percentile Rank 30 = 100 45

10

+ =

c) ( ) ( )3 0.5Percentile Rank 15 = 100 35

10

+ =

3.39 63 65 82 83 97 104 106 117 118 124 136 141

a)

( ) ( )

( ) ( )



Percentile Rank 100 = 100 45.812

+

+ =



b) ( ) ( )2 0.5Percentile Rank 75 = 100 20.8

12

+ =

c) ( ) ( )9 0.5Percentile Rank 120 = 100 79.2

12

+ =

3.40 12 14 14 15 16 18 18 20 20 22 23 24 26 a)

( ) ( )1

1

25Q : 13 3.25

100 10015

Pi n

Q

= = =

=

( ) ( )2

2

50Q : 13 6.5

100 10018

Pi n

Q

= = =

=

( ) ( )3

3

75Q : 13 9.75

100 10022

Pi n

Q

= = =

=

b) 3 1 = 22 15 7IQR Q Q− = − =

3.41 2 16 23 24 28 35 36 43 54 55 59 81

( ) ( )

( )1

1

25Q : 12 3

100 10023 24 / 2 23.5

Pi n

Q

= = =

= + =

( ) ( )

( )2

2

50Q : 12 6

100 10035 36 / 2 35.5

Pi n

Q

= = =

= + =

( ) ( )

( )3

3

75Q : 12 9

100 10054 55 / 2 54.5

Pi n

Q

= = =

= + =

The 5-number summary is 2, 23.5, 35.5, 54.5, 81 3.42 0.82 0.82 0.95 1.20 1.31 1.33 1.45 1.53 2.10 2.70 3.79 8.18

a) ( ) ( )6012 7.2

100 100

Pi n= = =

60th percentile = 1.53

3-24 Chapter 3


b) ( ) ( )2012 2.4

100 100

Pi n= = =


c) ( ) ( )8512 10.2

100 100

Pi n= = =


3.43 583 6.9 613 637 650 651 651 669 684 700 718 731 734 758 765 776

a)

( ) ( )

( ) ( )



Percentile Rank Mets = 100 28.116

+

+ =

b) ( ) ( )9 0.5Percentile Rank Braves = 100 59.4

16

+ =

c) ( ) ( )8 0.5Percentile Rank Phillies = 100 53.1

16

+ =

3.44 8 10 18 58 58 59 63 64 69 71 75 78 80 82 84 84 86 87 87 88 a)

( ) ( )

( )1

1

25Q : 20 5

100 10058 59 / 2 58.5

Pi n

Q

= = =

= + =

( ) ( )

( )2

2

50Q : 20 10

100 10071 75 / 2 73

Pi n

Q

= = =

= + =

( ) ( )

( )3

3

75Q : 20 15

100 10084 84 / 2 84

Pi n

Q

= = =

= + =

b) 3 1 = 84 58.5 25.5IRQ Q Q− = − =



3.45 170 170 171 171 172 174 174 177 179 181 185 189 191 193 193 195 195 197 198 199 204 208 212 217 223 225 238 242 261 269 a)

( ) ( )1

1

25Q : 30 7.5

100 100177

Pi n

Q

= = =

=

( ) ( )

( )2

2

50Q : 30 15

100 100193 195 / 2 194

Pi n

Q

= = =

= + =

( ) ( )3

3

75Q : 30 22.5

100 100212

Pi n

Q

= = =

=

3 1 = 212 177 35IQR Q Q− = − =

( ) ( )( ) ( )

3

1

Upper Limit 1.5 212 1.5 35 264.5

Lower Limit 1.5 177 1.5 35 124.5

Q IQR

Q IQR

= + = + =

= − = − =

The value 269 is an outlier.

b) The 5-number summary is 170, 177, 194, 212, 269

3-26 Chapter 3


3.46 a)

1 125 305.0 6.0

5 5

n n

i ii i

x yx y

n n= == = = = = =

ix x ( )ix x− iy y ( )iy y− ( )( )i ix x y y− −

4 5.0 –1.0 4 6.0 –2.0 2.0 7 5.0 2.0 9 6.0 3.0 6.0 2 5.0 –3.0 5 6.0 –1.0 3.0 6 5.0 1.0 5 6.0 –1.0 –1.0 6 5.0 1.0 7 6.0 1.0 1.0

( )( ) 11i ix x y y− − =

( )( )1 11

2.751 5 1

n

i ii

xy

x x y ys

n=

− −= = =

− −

b)

( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )( )

2

1

2 2 2 2 2

2 2 2 2 2

1

4 5 7 5 2 5 6 5 6 5 16= 2.0

5 1 4

4 6 9 6 5 6 5 6 7 6 16= 2.0

5 1 4

2.750.688

2.0 2.0

n

ii

x

x

y

xyxy

x y

x xs

n

s

s

sr

s s

=

−=

−

− + − + − + − + −= =

−

− + − + − + − + −= =

−

= = =

c) There is a positive linear relationship between these two variables.



3.47 a)

1 118 304.5 7.5

4 4

n n

i ii i

x yx y

n n= == = = = = =


7 4.5 2.5 6 7.5 –1.5 –3.75 6 4.5 1.5 8 7.5 0.5 0.75 2 4.5 –2.5 11 7.5 3.5 –8.75 3 4.5 –1.5 5 7.5 –2.5 3.75

( )( ) 8.0i ix x y y− − = −

( )( )1 8.0

2.671 4 1

n

i ii

xy

x x y ys

n=

− −−= = = −

− −

b)

( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )( )

2

1

2 2 2 2

2 2 2 2

1

7 4.5 6 4.5 2 4.5 3 4.5 17= 2.38

4 1 3

6 7.5 8 7.5 11 7.5 5 7.5 21= 2.65

4 1 3

2.670.423

2.38 2.65

n

ii

x

x

y

xyxy

x y

x xs

n

s

s

sr

s s

=

−=

−

− + − + − + −= =

−

− + − + − + −= =

−−= = = −

c) There is a negative linear relationship between these two variables.

3-28 Chapter 3


3.48 a) x = minutes, y = rating

1 150 1475.0 14.7

10 10

n n

i ii i

x yx y

n n= == = = = = =


4 5.0 –1.0 15 14.7 0.3 –0.3 8 5.0 3.0 13 14.7 –1.7 –5.1 0 5.0 –5.0 18 14.7 3.3 –16.5 5 5.0 0.0 10 14.7 –4.7 0.0 6 5.0 1.0 14 14.7 –0.7 –0.7 2 5.0 –3.0 16 14.7 1.3 –3.9 10 5.0 5.0 14 14.7 –0.7 –3.5 3 5.0 –2.0 20 14.7 5.3 –10.6 8 5.0 3.0 14 14.7 –0.7 –2.1 4 5.0 –1.0 13 14.7 –1.7 1.7

( )( ) 41.0i ix x y y− − = −

( )( )1 41.0

4.561 10 1

n

i ii

xy

x x y ys

n=

− −−= = = −

− −

b)

ix 2x iy 2y

4 16 15 225 8 64 13 169 0 0 18 324 5 25 10 100 6 36 14 196 2 4 16 256 10 100 14 196 3 9 20 400 8 64 14 196 4 16 13 169 Total 50 334 147 2,231



( ) ( )

2 2

2 21 1

1 1

2 2

1 1

50 147334 2, 231

10 10 10 1 10 1

3.06 2.79

n n

i in ni i

i ii i

x y

x y

x y

x yx y

n ns sn n

s s

s s

= =

= =

− −

= =− −

− −= =

− −= =

( )( )4.56

0.534 3.06 2.79

xyxy

x y

sr

s s−= = = −

c) There is a negative linear relationship between these two variables. 3.49 a) x = engine size, y = MPG

1 122.5 2252.5 25.0

9 9

n n

i ii i

x yx y

n n= == = = = = =


2.0 2.5 –0.5 26 25 1 –0.5 2.2 2.5 –0.3 30 25 5 –1.5 2.4 2.5 –0.1 25 25 0 0.0 3.3 2.5 0.8 21 25 –4 –3.2 3.0 2.5 0.5 23 25 –2 –1.0 2.0 2.5 –0.5 28 25 3 –1.5 2.4 2.5 –0.1 24 25 –1 0.1 2.0 2.5 –0.5 29 25 4 –2.0 3.2 2.5 0.7 19 25 –6 –4.2

( )( ) 13.8i ix x y y− − = −

( )( )1 13.8

1.731 9 1

n

i ii

xy

x x y ys

n=

− −−= = = −

− −

3-30 Chapter 3


b)

ix 2x iy 2y

2.0 4.00 26 676 2.2 4.48 30 900 2.4 5.76 25 625 3.3 10.89 21 441 3.0 9.00 23 529 2.0 4.00 28 784 2.4 5.76 24 576 2.0 4.00 29 841 3.2 10.24 19 361 Total 22.5 58.49 225 5,733

( ) ( )

2 2

2 21 1

1 1

2 2

1 1

22.5 22558.49 5,733

9 9 9 1 9 1

0.529 3.67

n n

i in ni i

i ii i

x y

x y

x y

x yx y

n ns sn n

s s

s s

= =

= =

− −

= =− −

− −= =

− −= =

( )( )1.73

0.891 0.529 3.67

xyxy

x y

sr

s s−= = = −

c) There is a negative linear relationship between these two variables.



3.50 a) x = shelf space, y = sales

1 136 634.0 7.0

9 9

n n

i ii i

x yx y

n n= == = = = = =


2 4 –2 3 7 –4 8 3 4 –1 3 7 –4 4 4 4 0 6 7 –1 0 6 4 2 5 7 –2 –4 5 4 1 18 7 11 11 2 4 –2 4 7 –3 6 4 4 0 5 7 –2 0 5 4 1 7 7 0 0 5 4 1 12 7 5 5

( )( ) 30i ix x y y− − =

( )( )1 30

3.751 9 1

n

i ii

xy

x x y ys

n=

− −= = =

− −

b)

ix 2x iy 2y

2 4 3 9 3 9 3 9 4 16 6 36 6 36 5 25 5 25 18 324 2 4 4 16 4 16 5 25 5 25 7 49 5 25 12 144 Total 36 160 63 637

3-32 Chapter 3


( ) ( )

2 2

2 21 1

1 1

2 2

1 1

36 63160 637

9 9 9 1 9 1

1.41 4.95

n n

i in ni i

i ii i

x y

x y

x y

x yx y

n ns sn n

s s

s s

= =

= =

− −

= =− −

− −= =

− −= =

( )( )3.75

0.537 1.41 4.95

xyxy

x y

sr

s s= = =

c) There is a positive linear relationship between these two variables. 3.51 a) x = shoe size, y = weight

1 183 828.3 8.2

10 10

n n

i ii i

x yx y

n n= == = = = = =


7.5 8.3 –0.8 7.6 8.2 –0.6 0.48 8.0 8.3 –0.3 7.8 8.2 –0.4 0.12 9.0 8.3 0.7 8.0 8.2 –0.2 –0.14 8.5 8.3 0.2 9.4 8.2 1.2 0.24 8.5 8.3 0.2 7.7 8.2 –0.5 –0.10 8.0 8.3 –0.3 6.9 8.2 –1.3 0.39 8.5 8.3 0.2 9.2 8.2 1.0 0.20 7.5 8.3 –0.8 6.5 8.2 –1.7 1.36 8.0 8.3 –0.3 9.1 8.2 0.9 –0.27 9.5 8.3 1.2 9.8 8.2 1.6 1.92

( )( ) 4.20i ix x y y− − =

( )( )1 4.20

0.4671 10 1

n

i ii

xy

x x y ys

n=

− −= = =

− −



b)

ix 2x iy 2y

7.5 56.25 7.6 57.76 8.0 64.00 7.8 60.84 9.0 81.00 8.0 64.00 8.5 72.25 9.4 88.36 8.5 72.25 7.7 59.29 8.0 64.00 6.9 47.61 8.5 72.25 9.2 84.64 7.5 56.25 6.5 42.25 8.0 64.00 9.1 82.81 9.5 90.25 9.8 69.04 Total 83.0 692.50 82.0 683.60

( ) ( )

2 2

2 21 1

1 1

2 2

1 1

83 82692.5 683.6

10 10 10 1 10 1

0.632 1.12

n n

i in ni i

i ii i

x y

x y

x y

x yx y

n ns sn n

s s

s s

= =

= =

− −

= =− −

− −= =

− −= =

( )( )0.467

0.660 0.632 1.12

xyxy

x y

sr

s s= = =

c) There is a positive linear relationship between these two variables. 3.52

( )( ) ( )( ) ( )( ) ( )( ) ( )( )5 7 4 7 3 5 3 4 2 7 104Ford: 6.1

5 4 3 3 2 17i i

i

w xx

w+ + + +

= = = =+ + + +

( )( ) ( )( ) ( )( ) ( )( ) ( )( )5 8 4 6 3 7 3 6 2 4 111Chevy: 6.5

5 4 3 3 2 17i i

i

w xx

w+ + + +

= = = =+ + + +

( )( ) ( )( ) ( )( ) ( )( ) ( )( )5 3 4 9 3 5 3 8 2 8 106Honda: 6.2

5 4 3 3 2 17i i

i

w xx

w+ + + +

= = = =+ + + +

Choose Chevy because it has the highest weighted average.

3-34 Chapter 3


3.53

a) 1 80 74 62 ... 47 79 1,18769.8

17 17

n

ii

xx

n= + + + + += = = =

b)

( ) ( )0.5 0.5 17 8.5i n= = =

13 44 45 47 55 56 62 64 74 74 79 79 80 87 88 98 142 Median = 74

c) The mode = 74, 79. d) Because 142 could be considered an outlier, the median would be the measurement that best describes the central tendency of these data. e) Because the mean is lower than the median, this distribution’s shape would be described as left-skewed. 3.54

a) 1 1.5 1.0 2.3 ... 1.5 12.4 93.010.33

9 9

n

ii

xx

n= + + + + += = = =

b)

( ) ( )0.5 0.5 9 4.5i n= = =

1.0 1.5 1.5 2.3 8.8 12.1 12.4 13.2 40.2 Median = 8.8

c) Mode = 1.5 d) Because 40.2 could be considered an outlier, the median would be the measurement that best describes the central tendency of these data. e) Because the mean is higher than the median, this distribution’s shape would be described as right-skewed. 3.55

a) 1 6 18 7 ... 16 13 16810.5

16 16

n

ii

xx

n= + + + + += = = =

b)

( ) ( )0.5 0.5 16 8i n= = =

5 6 6 6 7 8 8 10 11 12 12 13 14 16 16 18

( )Median 10 11 / 2 10.5= + =



c) Mode = 6 d) Because there does not appear to be any an outliers, the mean or the median would adequately describe the central tendency of these data. e) Because the mean equals the median, this distribution’s shape would be described as symmetrical. 3.56

a) $207.72x = b) Median $196.87=

c) There is no mode in this data set. d) Because there does not appear to be any an outliers, the mean or the median would adequately describe the central tendency of these data. e) Because the mean is more than the median, this distribution’s shape would be described as right-skewed. 3.57 a) Range 5 0 5= − = b)

ix 2ix

1 1 5 25 3 9 1 1 2 4 2 4 1 1 4 16 0 0 1 1

10

1

20ii

x=

= 10

2

1

62ii

x=

=

( )

2

22 1

2 1

2062

10 2.441 10 1

n

ini

ii

xx

nsn

=

=

− −

= = =− −

c) 2.44 1.56s = =

3-36 Chapter 3


3.58 a) Range 25.5 4.5 21.0= − =

b)

ix 2ix

7.4 54.76 10.7 114.49 25.5 650.25 7.9 62.41 4.5 20.25 8.1 65.61 5.4 29.16 15.0 225.00 5.3 28.09 7.7 59.29

10

1

97.5ii

x=

= 10

2

1

1,309.31ii

x=

=

( )

2

22 1

2 1

97.51,309.31

10 39.851 10 1

n

ini

ii

xx

nsn

=

=

− −

= = =− −

c) 39.85 6.31s = = 3.59 a) Range 22 7 15= − = b)

ix 2ix

17 289 22 484 8 64 19 361 20 400 21 441 7 49 14 196

8

1

128ii

x=

= 8

2

1

2, 284ii

x=

=



( )

2

22 1

2 1

1282, 284

8 33.711 8 1

n

ini

ii

xx

nsn

=

=

− −

= = =− −

c) 33.71 5.81s = =

3.60

a) ( )Range 5 15 20= − − =

b)

ix 2ix

-12 144 -8 64 -15 225 -10 100 4 16 -6 36 -9 81 5 25 -8 64 -10 100

10

1

69ii

x=

= − 10

2

1

855ii

x=

=

( )

2

22 1

2 1

69855 855 476.110 42.1

1 10 1 9

n

ini

ii

xx

nsn

=

=

− − − −= = = =− −

c) 42.1 6.49s = =

3.61 a) Range 1,727= b) 2 162,609.9s = c) 403.25s =

3-38 Chapter 3


3.62 a) Store 1 Store 2

ix 2ix ix 2

ix

7 49 8 64 4 16 7 49 6 36 10 100 3 9 8 64 9 81 8 64

5

1

29ii

x=

= 5

2

1

191ii

x=

=

5

1

41ii

x=

= 5

2

1

341ii

x=

=

Store 1: 1 29

5.85

n

ii

xx

n== = =

Store 2:

1 418.2

5

n

ii

xx

n== = =

Store 1:

( )

2

22 1

1

29191 191 168.25 2.39

1 5 1 4

n

ini

ii

xx

nsn

=

=

− − −= = = =− −

Store 2:

( )241

341 341 336.25 1.105 1 4

s− −= = =

−

Store 1: ( ) ( )2.39100 100 41.2%

5.8

sCVx

= = =

Store 2: ( )1.10100 13.4%

8.2CV = =

b) Store 1 has more variability because it has a higher coefficient of variation.



3.63 New England Detroit

ix 2ix ix 2

ix

14 196 6 36 10 100 5 25 12 144 3 9 16 256 7 49 11 121 0 0 10 100 2 4 14 196 6 36 13 169 10 100 12 144 4 16

9

1

112ii

x=

= 9

2

1

1, 426ii

x=

=

9

1

43ii

x=

= 9

2

1

275ii

x=

=

NE: 1 112

12.49

n

ii

xx

n== = =

Det:

1 434.8

9

n

ii

xx

n== = =

NE:

( )

2

22 1

1

1121, 426

9 2.011 9 1

n

ini

ii

xx

nsn

=

=

− −

= = =− −

Det:

( )243

2759 2.95

9 1s

−= =

−

NE: ( ) ( )2.01100 100 16.2%

12.4

sCVx

= = =

Det: ( )2.95100 61.5%

4.8CV = =

Detroit has more variability in games won because they have a higher coefficient of variation.

3-40 Chapter 3


3.64

ix 2ix ix 2

ix

423 178,929 210 44,100 407 165,649 203 41.209 264 69,696 202 40,804 252 63,504 189 35,721 245 60,025 186 34,596 236 55,696 172 29,584 233 54,289 165 27,225 227 51,529 156 24,336 227 51,529 131 17,161 211 44,521 130 16,900

20

1

4, 469ii

x=

= 20

2

1

1,107,003ii

x=

=

1 $4,469$223.45

20

n

ii

xx

n== = =

( )

2

22 1

1

4, 4691,107,003

20 $75.531 20 1

n

ini

ii

xx

nsn

=

=

− −

= = =− −

a) ( ) ( )$75.53100 100 32.4%

$223.45

sCVx

= = =

b) $200 $223.45

0.31$75.53

x xzs− −= = = −

c)

( )( )( )( )

$223.45 1 $75.53 $298.98

$223.45 1 $75.53 $147.92

x x zsx

x

= += + =

= − = 16/20 = 80% of the values are within this interval. This exceeds the 68% predicted by the Empirical Rule. d)

( )( )( ) ( )

$223.45 2 $75.53 $374.51

$223.45 2 $75.53 $72.39

x x zsx

x

= += + =

= − =

18/20 = 90% of the values are within this interval. This exceeds the 75% predicted by Chebyshev’s Theorem.



3.65 a) x = missed classes, y = final grade

1 127 7743.0 86.0

9 9

n n

i ii i

x yx y

n n= == = = = = =


4 3 1 81 86 –5 –5 7 3 4 79 86 –7 –28 2 3 –1 93 86 7 –7 5 3 2 70 86 –16 –32 0 3 –3 96 86 10 –30 2 3 –1 84 86 –2 2 0 3 –3 90 86 4 –12 5 3 2 86 86 0 0 2 3 –1 95 86 9 –9

( )( ) 121i ix x y y− − = −

( )( )1 121

15.131 9 1

n

i ii

xy

x x y ys

n=

− −−= = = −

− −

b)

ix 2x iy 2y

4 16 81 6,561 7 49 79 6,241 2 4 93 8,649 5 25 70 4,900 0 0 96 9,216 2 4 84 7,056 0 0 90 8,100 5 25 86 7,396 2 4 95 9,025 Total 27 127 774 67,144

3-42 Chapter 3


( ) ( )

2 2

2 21 1

1 1

2 2

1 1

27 774127 67,144

9 9 9 1 9 1

2.40 8.51

n n

i in ni i

i ii i

x y

x y

x y

x yx y

n ns sn n

s s

s s

= =

= =

− −

= =− −

− −= =

− −= =

( )( )15.13

0.741 2.40 8.51

xyxy

x y

sr

s s−= = = −

c) There is a negative linear relationship between these two variables. 3.66 a) x = satisfaction score, y = activations

1 162.4 2447.8 30.5

8 8

n n

i ii i

x yx y

n n= == = = = = =


8 7.8 0.2 36 30.5 5.5 1.10 7.9 7.8 0.1 25 30.5 –5.5 –0.55 8.5 7.8 0.7 40 30.5 9.5 6.65 9.0 7.8 1.2 38 30.5 7.5 9.00 6.1 7.8 –1.7 19 30.5 –11.5 19.55 7.0 7.8 –0.8 28 30.5 –2.5 2.00 8.2 7.8 0.4 33 30.5 2.5 1.00 7.7 7.8 –0.1 25 30.5 –5.5 0.55

( )( ) 39.30i ix x y y− − =

( )( )1 39.30

5.611 8 1

n

i ii

xy

x x y ys

n=

− −= = =

− −



b)

ix 2x iy 2y

8.0 64.00 36 1,296 7.9 62.41 25 625 8.5 72.25 40 1,600 9.0 81.00 38 1,444 6.1 37.21 19 361 7.0 49.00 28 784 8.2 67.24 33 1,089 7.7 59.29 25 625 Total 62.4 492.4 244 7,824

( ) ( )

2 2

2 21 1

1 1

2 2

1 1

62.4 244492.4 7,824

8 8 8 1 8 1

0.901 7.39

n n

i in ni i

i ii i

x y

x y

x y

x yx y

n ns sn n

s s

s s

= =

= =

− −

= =− −

− −= =

− −= =

( )( )5.61

0.843 0.901 7.39

xyxy

x y

sr

s s= = =

c) There is a positive linear relationship between these two variables. 3.67

Using Excel, 904.30x = and 46.78s = .

a) ( ) ( )46.78100 100 5.2%

904.30

sCVx

= = =

b) 900 904.3

0.0946.78

x xzs− −= = = −

c)

( )( )( )( )

904.3 2 46.78 997.86

904.3 2 46.78 810.74

x x zsx

x

= += + =

= − = 25/25 = 100% of the values are within this interval. This exceeds the 95% predicted by the Empirical Rule.

3-44 Chapter 3


d)

( )( )( )( )

904.3 3 46.78 1,044.64

904.3 3 46.78 763.96

x x zsx

x

= += + =

= − =

25/25 = 100% of the values are within this interval. This exceeds the 89% predicted by Chebyshev’s Theorem. 3.68 Scores Midpoint 278 to 282 280 283 to 287 285 288 to 292 290 293 to 297 295 298 to 302 300 303 to 307 305 a)

( )

( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )6 280 12 285 21 290 16 295 6 300 1 305 18,015290.6

6 12 21 16 6 1 62

i if mN

μ

μ

≈

+ + + + +≈ = =

+ + + + +

Value mi fi μ ( )μ−im ( )μ− 2

im ( )μ− 2

i im f

278 to 282 280 6 290.6 –10.6 112.36 674.16 283 to 287 285 12 290.6 –5.6 31.36 376.32 288 to 292 290 21 290.6 –0.6 0.36 7.56 293 to 297 295 16 290.6 4.4 19.36 309.76 298 to 302 300 6 290.6 9.4 88.36 530.16 303 to 307 305 1 290.6 14.4 207.36 207.36

62iN f= = ( )22,105.32i im fμ− =

b)

( )2

1 2,105.325.83

62

k

i ii

m f

N

μσ =

−≈ = =



3.69 Mileage Midpoint 5,000 to under 6,000 5,500 6,000 to under 7,000 6,500 7,000 to under 8,000 7,500 8,000 to under 9,000 8,500 9,000 to under 10,000 9,500 10,000 to under 11,000 10,500 a) All values are in thousands of miles.

( )( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )16 5.5 11 6.5 24 7.5 10 8.5 15 9.5 9 10.5 661.57.78 7,780 miles

16 11 24 10 15 9 85

i if xx

n

x

=

+ + + + += = = =

+ + + + +


im x ( )−2

i im x f

5 to under 6 5.5 16 7.78 -2.28 5.1984 83.1744 6 to under 7 6.5 11 7.78 -1.28 1.6384 18.0224 7 to under 8 7.5 24 7.78 -0.28 0.0784 1.8816 8 to under 9 8.5 10 7.78 0.72 0.5184 5.1840 9 to under 10 9.5 15 7.78 1.72 2.9584 44.3760 10 to under 11 10.5 9 7.78 2.72 7.3984 66.5856

85= = in f ( )2219.2240− = i im x f

b)

( )2

1 219.2241.615 1,615 miles

1 85 1

k

i ii

m x fs

n=

−≈ = = =

− −

3.70 $7.75 $8.40 $9.50 $10.00 $10.35 $10.50 $10.75 $11.00 $11.45 $12.15 $12.40 $12.75 $13.75 $14.00 $14.10 $15.50 $16.00 $16.50 $17.60 $17.85 $18.40

a) ( ) ( )3021 6.3

100 100

Pi n= = =

30th percentile = $10.75

b) ( ) ( )4521 9.45

100 100

Pi n= = =


3-46 Chapter 3


c) ( ) ( )6521 13.65

100 100

Pi n= = =


3.71

13 19 20 21 24 24 25 26 28 29

30 30 32 33 34 40 42 42 45 46 a)

( ) ( )

( )

2020 4

100 10020th percentile 21 24 / 2 22.5

Pi n= = =

= + =

b)

( ) ( )

( )

7020 14

100 10070th percentile 33 34 / 2 33.5

Pi n= = =

= + =

c)

( ) ( )

( )

9020 18

100 10090th percentile 42 45 / 2 43.5

Pi n= = =

= + =

3.72 387 454 490 564 629 665 668 673 680 696 815 845 894 919 941 974 1,040 1,074 1,165 1,175 1,188 1,245 1,345 1,419 1,471 1,535 1,554 a)

( ) ( )1

1

25Q : 27 6.75

100 100668

Pi n

Q

= = =

=

( ) ( )2

2

50Q : 27 13.5

100 100919

Pi n

Q

= = =

=

( ) ( )3

3

75Q : 27 20.25

100 1001,188

Pi n

Q

= = =

=

b)

3 1 = 1,188 668 520IQR Q Q− = − =



3.73 84 86 94 103 104 106 114 115 121 122 123 124 138 146 147 153 156 157 161 166 168 178 183 183 189 201 202 202 222 227 a)

( ) ( )1

1

25Q : 30 7.5

100 100115

Pi n

Q

= = =

=

( ) ( )

( )2

2

50Q : 30 15

100 100147 153 / 2 150

Pi n

Q

= = =

= + =

( ) ( )3

3

75Q : 30 22.5

100 100183

Pi n

Q

= = =

=

b)

3 1 = 183 115 68IQR Q Q− = − =

3.74

( ) ( )

( ) ( )

( ) ( )



Percentile Rank Freestyle 33 seconds = 100 68.882 0.5

Percentile Rank Backstroke 72 seconds = 100 25.010

+

+ =

+ =

Because lower times are a measure of better performance, the swimmer performed better in the backstroke relative to the freestyle competition. 3.75

66 69 70 72 73 76 78 78 79 79 80 80 81 84 84 85 86 87 88 89 89 92 96 99 100

a)

( ) ( )

( ) ( )



Percentile Rank 70 = 100 1025

+

+ =

b) ( ) ( )5 0.5Percentile Rank 75 = 100 22

25

+ =

3-48 Chapter 3


c) ( ) ( )13 0.5Percentile Rank 82 = 100 54

25

+ =

d) ( ) ( )21 0.5Percentile Rank 90 = 100 86

25

+ =

3.76 215 219 229 230 236 239 240 244 247 255 262 264 271 279 280 282 285 285 290 296 301 310 326 327 330 a)

( ) ( )1

1

25: 25 6.25

100 100240

PQ i n

Q

= = =

=

( ) ( )2

2

50Q : 25 12.5

100 100271

Pi n

Q

= = =

=

( ) ( )3

3

75Q : 25 18.75

100 100290

Pi n

Q

= = =

=

3 1 = 290 240 50IQR Q Q− = − =

( ) ( )( ) ( )

3

1

Upper Limit 1.5 290 1.5 50 365

Lower Limit 1.5 240 1.5 50 165

Q IQR

Q IQR

= + = + =

= − = − =

There are no outliers in this data set.

b) The 5-number summary is 215, 240, 271, 290, 330 3.77 59.8 61.0 63.0 67.2 69.7 69.9 70.7 75.1 75.2 76.8 80.9 81.5 82.8 83.5 83.6 86.4 86.8 88.9 92.9 93.1 93.2 96.2 97.6 98.6 99.9 100.5 103.2 104.4 107.2 109.6



a)

( ) ( )1

1

25Q : 30 7.5

100 10075.1

Pi n

Q

= = =

=

( ) ( )

( )2

2

50Q : 30 15

100 10083.6 86.4 / 2 85.0

Pi n

Q

= = =

= + =

( ) ( )3

3

75Q : 30 22.5

100 10097.6

Pi n

Q

= = =

=

3 1 = 97.6 75.1 22.5IQR Q Q− = − =

( ) ( )( ) ( )

3

1

Upper Limit 1.5 97.6 1.5 22.5 131.4

Lower Limit 1.5 75.1 1.5 22.5 41.4

Q IQR

Q IQR

= + = + =

= − = − =

There are no outliers in this data set.

b) The five-number summary is 59.8, 75.1, 85.0, 97.6, 109.6 3.78 a) x = age, y = credit score

1 1400 6,93040 693

10 10

n n

i ii i

x yx y

n n= == = = = = =


36 40 –4 675 693 –18 72 24 40 –16 655 693 –38 608 54 40 14 760 693 67 938 28 40 –12 615 693 –78 936 31 40 –9 660 693 –33 297 47 40 7 790 693 97 679 35 40 –5 720 693 27 –135 59 40 19 760 693 67 1,273 40 40 0 685 693 –8 0 46 40 6 610 693 –83 –498

( )( ) 4,170i ix x y y− − =

3-50 Chapter 3


( )( )1 4,170

463.331 10 1

n

i ii

xy

x x y ys

n=

− −= = =

− −

b)

ix 2x iy 2y

36 1,296 675 455,625 24 576 655 429,025 54 2,916 760 577,600 28 784 615 378,225 31 961 660 435,600 47 2,209 790 624,100 35 1,225 720 518,400 59 3,481 760 577,600 40 1,600 685 469,225 46 2,116 610 372,100

Total 400 17,164 6,930 4,837,500

( ) ( )

2 2

2 21 1

1 1

2 2

1 1

400 6,93017,164 4,837,500

10 10 10 1 10 1

11.37 62.37

n n

i in ni i

i ii i

x y

x y

x y

x yx y

n ns sn n

s s

s s

= =

= =

− −

= =− −

− −= =

− −= =

( )( )463.33

0.653 11.37 62.37

xyxy

x y

sr

s s= = =

c) There is a positive linear relationship between these two variables.



3.79 a) x = screen size, y = battery life

1 1112 3214.0 4.0

8 8

n n

i ii i

x yx y

n n= == = = = = =


15.6 14.0 1.6 3.6 4.0 –0.4 –0.64 17.3 14.0 3.3 4.5 4.0 0.5 1.65 14.5 14.0 0.5 4.2 4.0 0.2 0.10 11.6 14.0 –2.4 4.5 4.0 0.5 –1.20 14.5 14.0 0.5 3.1 4.0 –0.9 –0.45 13.3 14.0 –0.7 3.4 4.0 –0.6 0.42 11.6 14.0 –2.4 4.8 4.0 0.8 –1.92 13.6 14.0 –0.4 3.9 4.0 –0.1 0.04

( )( ) 2.00i ix x y y− − = −

( )( )1 2.00

0.2861 8 1

n

i ii

xy

x x y ys

n=

− −−= = = −

− −

b)

ix 2x iy 2y

15.6 243.36 3.6 12.96 17.3 299.29 4.5 20.25 14.5 210.25 4.2 17.64 11.6 134.56 4.5 20.25 14.5 210.25 3.1 9.61 13.3 176.89 3.4 11.56 11.6 134.56 4.8 23.04 13.6 184.96 3.9 15.21 Total 112.0 1,594.12 32.0 130.52

( ) ( )

2 2

2 21 1

1 1

2 2

1 1

112 321,594.12 130.52

8 8 8 1 8 1

1.93 0.600

n n

i in ni i

i ii i

x y

x y

x y

x yx y

n ns sn n

s s

s s

= =

= =

− −

= =− −

− −= =

− −= =

3-52 Chapter 3


( )( )0.286

0.247 1.93 0.600

xyxy

x y

sr

s s−= = = −

c) There is a weak negative linear relationship between these two variables.

CHAPTER 3 Calculating Descriptive Statistics 3 Calculating Descriptive Statistics 3.1 a) 1 18 12 7...

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Transcript of CHAPTER 3 Calculating Descriptive Statistics 3 Calculating Descriptive Statistics 3.1 a) 1 18 12 7...