Manual de solucoes para o professor em ingles - Cap3

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Chapter 3 Descriptive Statistics: Numerical Measures Learning Objectives 1.

Understand the purpose of measures of location.

2.

Be able to compute the mean, median, mode, quartiles, and various percentiles.

3.

Understand the purpose of measures of variability.

4.

Be able to compute the range, interquartile range, variance, standard deviation, and coefficient of variation.

5.

Understand skewness as a measure of the shape of a data distribution. Learn how to recognize when a data distribution is negatively skewed, roughly symmetric, and positively skewed.

6.

Understand how z scores are computed and how they are used as a measure of relative location of a data value.

7.

Know how Chebyshev’s theorem and the empirical rule can be used to determine the percentage of the data within a specified number of standard deviations from the mean.

8.

Learn how to construct a 5-number summary and a box plot.

9.

Be able to compute and interpret covariance and correlation as measures of association between two variables.

10.

Be able to compute a weighted mean.

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Chapter 3 Solutions:

x

1.

xi 75   15 n 5

10, 12, 16, 17, 20 Median = 16 (middle value)

x

2.

xi 96   16 n 6

10, 12, 16, 17, 20, 21 Median = 3.

15, 20, 25, 25, 27, 28, 30, 34

i

20 (8)  1.6 100

2nd position = 20

i

25 (8)  2 100

20  25  22.5 2

i

65 (8)  5.2 100

6th position = 28

i

75 (8)  6 100

28  30  29 2

Mean 

4.

5.

16  17  16.5 2

xi 657   59.73 n 11

Median = 57

6th item

Mode = 53

It appears 3 times

xi 3181   $159 n 20

a.

x

b.

Median 10th $160 11th $162 Median =

c.

Los Angeles Seattle

160  162  $161 2

Mode = $167 San Francisco and New Orleans

3-2 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part

Descriptive Statistics: Numerical Measures d.

 25  i  20  5  100  5th 6th

Q1 

e.

$134 $139

134  139  $136.50 2

 75  i  20  15  100  15th $167 16th $173

Q3 

6.

167  173  $170 2

a.

x

xi 350   18.42 n 19

b.

x

xi 120   6.32 n 19

c.

120 (100)  34.3% of 3-point shots were made from the 20 feet, 9 inch line during the 19 games. 350

d.

Moving the 3-point line back to 20 feet, 9 inches has reduced the number of 3-point shots taken per game from 19.07 to 18.42, or 19.07 – 18.42 = .65 shots per game. The percentage of 3-points made per game has been reduced from 35.2% to 34.3%, or only .9%. The move has reduced both the number of shots taken per game and the percentage of shots made per game, but the differences are small. The data support the Associated Press Sports conclusion that the move has not changed the game dramatically. The 2008-09 sample data shows 120 3-point baskets in the 19 games. Thus, the mean number of points scored from the 3-point line is 120(3)/19 = 18.95 points per game. With the previous 3-point line at 19 feet, 9 inches, 19.07 shots per game and a 35.2% success rate indicate that the mean number of points scored from the 3-point line was 19.07(.352)(3) = 20.14 points per game. There is only a mean of 20.14 – 18.95 = 1.19 points per game less being scored from the 20 feet, 9 inch 3point line.

3-3 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part. .

Chapter 3

7.

xi 148   14.8 n 10

a.

x

b.

Order the data from low 6.7 to high 36.6 Median

 50  th th i 10  5 Use 5 and 6 positions. 100  

Median 

10.1  16.1  13.1 2

c.

Mode = 7.2 (occurs 2 times)

d.

 25  i 10  2.5 Use 3rd position. Q1 = 7.2   100 

 75  i 10  7.5  100  e.

Use 8th position. Q3 = 17.2

Σxi = $148 billion The percentage of total endowments held by these 2.3% of colleges and universities is (148/413)(100) = 35.8%.

f.

8.

a.

A decline of 23% would be a decline of .23(148) = $34 billion for these 10 colleges and universities. With this decline, administrators might consider budget cutting strategies such as    

Hiring freezes for faculty and staff Delaying or eliminating construction projects Raising tuition Increasing enrollments

x

x

i

n



3200  160 20

Order the data from low 100 to high 360 Median

 50  th th i  20  10 Use 10 and 11 positions  100 

 130  140  Median =    135 2   Mode = 120 (occurs 3 times)

3-4 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part

Descriptive Statistics: Numerical Measures b.

 25  th th i  20  5 Use 5 and 6 positions  100   115  115  Q1     115 2    75  th th i  20  15 Use 15 and 16 positions  100   180  195  Q3     187.5 2  

c.

 90  th th i  20  18 Use 18 and 19 positions  100   235  255  90th percentile     245 2   90% of the tax returns cost $245 or less. 10% of the tax returns cost $245 or more.

9.

a.

Ordered data: 112.8 140.2 169.9

177.5

181.3 202.5 230.0 315.5 470.2

With n = 9, the median is the 5th position. The median sales price of existing homes is $181.3 thousand. b.

Ordered data: 149.5

175.0

195.8

215.5

225.3

275.9

350.2

525.0

With n = 8, the median is the average of the 4th and 5th positions. The median sales price of new homes =

215.5  225.3  $220.4 thousand. 2

c.

New homes have the higher median sale price by $220.4 – 181.3 = $39.1 thousand

d.

Existing homes:

New homes:

181.3  208.4 27.1   .130 or a 13.0% decrease in the median sales price. 208.4 208.4

220.4  249.0 28.6   .115 or an 11.5% decrease in the median sales price. 249.0 249.0

Existing homes had the larger one-year percentage decrease in the median sales price. However, new homes have had the larger one-year decrease in the median sales price; a median sales price decrease of $28.6 thousand for new homes and a median sales price decrease of $27.1 thousand for existing homes.

3-5 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part. .

Chapter 3 10. a. b.

Minimum = .4%; Maximum = 3.5%

xi = 69 x

xi 69   2.3% n 30

Median is average of 15th and 16th items. Both are 2.5%, so the median is 2.5%. The mode is 2.7%; forecast by 4 economists. c.

For Q1,

 25  i 30  7.5; round up and use the 8th item   100  Q1 = 2.0% For Q3,

 75  i  30  22.5; round up and use the 23rd item  100  Q3 = 2.8% d. 11.

Generally, the 2% to 3% growth should be considered optimistic. Using the mean we get xcity =15.58,

xhighway = 18.92

For the samples we see that the mean mileage is better on the highway than in the city. City 13.2 14.4 15.2 15.3 15.3 15.3 15.9 16 16.1 16.2 16.2 16.7 16.8  Median Mode: 15.3 Highway 17.2 17.4 18.3 18.5 18.6 18.6 18.7 19.0 19.2 19.4 19.4 20.6 21.1  Median Mode: 18.6, 19.4 The median and modal mileages are also better on the highway than in the city.

3-6 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part

Descriptive Statistics: Numerical Measures 12.

Disney Total Revenue: $3,321 million (13 movies)

x

xi 3321   $255.5 n 13

104 110 136 169 249 250 253 273 304 325 346 354 448 Median 7th position Median = $253 4th position

Q1: i = .25(13) = 3.25 Q1 = $169

10th position

Q3: i = .75(13) = 9.75 Q3 = $325 Pixar

Total Revenue: $3,231 million (6 movies)

x

xi 3231   $538.5 n 6

362 363 485 525 631 865 Median (3rd and 4th positions) Median =

485  525  $505 2

Q1: i = .25(6) = 1.5

2nd position

Q1 = $363 Q3: i = .75(6) = 4.5

5th position

Q3 = $631 The total box office revenues for the two companies over the 10 year period are approximately the same: Disney $3321 million; Pixar $3231 million. But Disney generated its revenue with 13 films while Pixar generated its revenue with only 6 films. Mean Median

Disney $225.5 $253

Pixar $538.5 $505

3-7 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part. .

Chapter 3 The first quartiles show 75% of Disney films do better than $169 million while 75% of Pixar films do better than $363 million. The third quartiles show 25% of Disney films do better than $325 million while 25% of Pixar films do better than $631. In all of these comparisons, Pixar films are about twice as successful as Disney films when it comes to box office revenue. In buying Pixar, Disney looks to acquire Pixar’s ability to make higher revenue films. 13.

Range 20 - 10 = 10 10, 12, 16, 17, 20

i

25 (5)  1.25 100

Q1 (2nd position) = 12

i

75 (5)  3.75 100

Q3 (4th position) = 17 IQR = Q3 - Q1 = 17 - 12 = 5 14.

x

s2 

xi 75   15 n 5 ( xi  x ) 2 64   16 n 1 4

s  16  4 15.

15, 20, 25, 25, 27, 28, 30, 34

Range = 34 - 15 = 19

i

25 (8)  2 100

Q1 

20  25  22.5 2

i

75 (8)  6 100

Q3 

28  30  29 2

IQR = Q3 - Q1 = 29 - 22.5 = 6.5

x

s2 

xi 204   255 . n 8 ( xi  x ) 2 242   34.57 n 1 7

s  34.57  588 .

3-8 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part

Descriptive Statistics: Numerical Measures 16. a. b.

Range = 190 - 168 = 22

( xi  x )2  376

s = 376 = 75.2 5 2

c.

s  752 .  8.67

d.

 8.67  Coefficient of Variation   100%  4.87%  178 

17. a.

b.

With DVD

x

xi 2050   410 n 5

Without DVD

x

xi 1550   310 n 5

With DVD

$410 - $310 = $100 more expensive

With DVD

Range = 500 - 300 = 200

s2 

( xi  x )2 22000   5500 n 1 4

s  5500  74.2 Without DVD

Range = 360 - 290 = 70

s2 

( xi  x )2 3200   800 n 1 4

s  800  28.3 Models with DVD players have the greater variation in prices. The price range is $300 to $500. Models without a DVD player have less variation in prices. The price range is $290 to $360. 18. a.

x

xi 266   $38 per day n 7

s2 

( xi  x )2 582   97 n 1 6

s  97  $9.85 b.

The mean car-rental rate per day is $38 for both Eastern and Western cities. However, Eastern cities show a greater variation in rates per day. This greater variation is most likely due to the inclusion of the most expensive city (New York) in the Eastern city sample.

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Chapter 3 19. a.

Range = 60 - 28 = 32 IQR = Q3 - Q1 = 55 - 45 = 10

b.

x

435  48.33 9

( xi  x )2  742 s2 

( xi  x )2 742   92.75 n 1 8

s  92.75  9.63 c. 20.

The average air quality is about the same. But, the variability is greater in Anaheim. Dawson Supply: Range = 11 - 9 = 2

4.1  .67 9 J.C. Clark: Range = 15 - 7 = 8 s

s 21. a.

60.1  2.58 9

Cities:

x

xi 198   $33 n 6

s2 

( xi  x )2 72   14.40 n 1 6 1

s  14.40  3.79 Retirement Areas:

x

xi 192   $32 n 6

s2 

( xi  x )2 18   3.60 n 1 6 1

s  3.60  1.90 b.

Mean cost of the market basket is roughly the same with the retirement areas sample mean $1 less. However, there is more variation in the cost in cities than in retirement areas.

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Descriptive Statistics: Numerical Measures 22. a.

Freshmen x 

Seniors

x

xi 32125   1285 n 25

xi 8660   $433 n 20

Freshmen spend almost three times as much on back-to-school items as seniors. b.

Freshmen Range = 2094 – 374 = 1720 Seniors

c.

Range = 632 – 280 = 352

Freshmen

 25  i  25  6.25  100 

Q1 = 1079 (7th item)

 75  i  25  18.75  100 

Q3 = 1475 (19th item)

IQR = Q3 - Q1 = 1479 – 1075 = 404 Seniors

 25  i  20  5  100 

Q1 

368  373  370.5 2

 75  i  20  15  100 

Q1 

489  515  502 2

IQR = Q3 - Q1 = 502 – 370.5 = 131.5 d.

s

( xi  x )2 n 1

Freshmen s 

Seniors e.

s

3233186  367.04 24 178610  96.96 19

All measures of variability show freshmen have more variation in back-to-school expenditures.

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Chapter 3 23. a.

For 2005

x

s

xi 608   76 n 8

( xi  x )2 30   2.07 n 1 7

For 2006

x

s b.

24.

xi 608   76 n 8

( xi  x )2 194   5.26 n 1 7

The mean score is 76 for both years, but there is an increase in the standard deviation for the scores in 2006. The golfer is not as consistent in 2006 and shows a sizeable increase in the variation with golf scores ranging from 71 to 85. The increase in variation might be explained by the golfer trying to change or modify the golf swing. In general, a loss of consistency and an increase in the standard deviation could be viewed as a poorer performance in 2006. The optimism in 2006 is that three of the eight scores were better than any score reported for 2005. If the golfer can work for consistency, eliminate the high score rounds, and reduce the standard deviation, golf scores should show improvement. Quarter milers s = 0.0564 Coefficient of Variation = (s/ x )100% = (0.0564/0.966)100% = 5.8% Milers s = 0.1295 Coefficient of Variation = (s/ x )100% = (0.1295/4.534)100% = 2.9% Yes; the coefficient of variation shows that as a percentage of the mean the quarter milers’ times show more variability.

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Descriptive Statistics: Numerical Measures

xi 75   15 n 5

x

25.

s2 

( xi  x ) 2  n 1

10

z

10  15  1.25 4

20

z

20  15  1.25 4

12

z

12  15  .75 4

17

z

16

64 4 4

17  15  .50 4 16  15 z  .25 4

z

520  500  .20 100

z

650  500  1.50 100

z

500  500  0.00 100

z

450  500  .50 100

z

280  500  2.20 100

27. a.

z

40  30 2 5

1

1  .75 At least 75% 22

b.

z

45  30 3 5

1

1  .89 At least 89% 32

c.

z

38  30  1.6 5

1

1  .61 At least 61% 1.62

d.

z

26.

e.

42  30  2.4 5 48  30 z  3.6 5

1  .83 At least 83% 2.42 1 1  .92 At least 92% 3.62 1

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Chapter 3

28. a.

Approximately 95%

b.

Almost all

c.

Approximately 68%

29. a.

This is from 2 standard deviations below the mean to 2 standard deviations above the mean. With z = 2, Chebyshev’s theorem gives:

1

1 1 1 3  1 2  1  z2 2 4 4

Therefore, at least 75% of adults sleep between 4.5 and 9.3 hours per day. b.

This is from 2.5 standard deviations below the mean to 2.5 standard deviations above the mean. With z = 2.5, Chebyshev’s theorem gives:

1

1 1 1  1  1  .84 2 2 6.25 z 2.5

Therefore, at least 84% of adults sleep between 3.9 and 9.9 hours per day. c.

With z = 2, the empirical rule suggests that 95% of adults sleep between 4.5and 9.3 hours per day. The percentage obtained using the empirical rule is greater than the percentage obtained using Chebyshev’s theorem.

30. a.

$1.95 is one standard deviation below the mean and $2.15 is one standard deviation above the mean. The empirical rule says that approximately 68% of gasoline sales are in the price range.

b.

Part (a) shows that approximately 68% of the gasoline sales are between $1.95 and $2.15. Since the bell-shaped distribution is symmetric, approximately half of 68%, or 34%, of the gasoline sales should be between $1.95 and the mean price of $2.05. $2.25 is two standard deviations above the mean price of $2.05. The empirical rule says that approximately 95% of the gasoline sales should be within two standard deviations of the mean. Thus, approximately half of 95%, or 47.5%, of the gasoline sales should be between the mean price of $2.05 and $2.25. The percentage of gasoline sales between $1.95 and $2.25 should be approximately 34% + 47.5% = 81.5%.

c.

$2.25 is two standard deviations above the mean and the empirical rule says that approximately 95% of the gasoline sales should be within two standard deviations of the mean. Thus, 1 - 95% = 5% of the gasoline sales should be more than two standard deviations from the mean. Since the bell-shaped distribution is symmetric, we expected half of 5%, or 2.5%, would be more than $2.25.

31. a.

615 is one standard deviation above the mean. Approximately 68% of the scores are between 415 and 615 with half of 68%, or 34%, of the scores between the mean of 515 and 615. Also, since the distribution is symmetric, 50% of the scores are above the mean of 515. With 50% of the scores above 515 and with 34% of the scores between 515 and 615, 50% - 34% = 16% of the scores are above 615.

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Descriptive Statistics: Numerical Measures b.

715 is two standard deviations above the mean. Approximately 95% of the scores are between 315 and 715 with half of 95%, or 47.5%, of the scores between the mean of 515 and 715. Also, since the distribution is symmetric, 50% of the scores are above the mean of 515. With 50% of the scores above 515 and with 47.5% of the scores between 515 and 715, 50%- 47.5% = 2.5% of the scores are above 715.

c.

Approximately 68% of the scores are between 415 and 615 with half of 68%, or 34%, of the scores between 415 and the mean of 515.

d.

Approximately 95% of the scores are between 315 and 715 with half of 95%, or 47.5%, of the scores between 315 and the mean of 515. Approximately 68% of the scores are between 415 and 615 with half of 68%, or 34%, of the scores between the mean of 515 and 615. Thus, 47.5% + 34% = 81.5% of the scores are between 315 and 615.

x



2300  3100  .67 1200



4900  3100  1.50 1200

32. a.

z

b.

z

c.

$2300 is .67 standard deviations below the mean. $4900 is 1.50 standard deviations above the mean. Neither is an outlier.

d.

z

 x



x





13000  3100  8.25 1200

$13,000 is 8.25 standard deviations above the mean. This cost is an outlier. 33. a.

x

xi 64   9.14 days n 7

Median: with n = 7, use 4th position 2, 3, 8, 8, 12, 13, 18 Median = 8 days Mode: 8 days (occurred twice)

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Chapter 3 b.

Range = Largest value – Smallest value = 18 – 2 = 16

s

( xi  x ) 2 n 1

( xi  x ) 2  (13  9.14) 2  (12  9.14) 2  (8  9.14) 2  (3  9.14) 2  (8  9.14) 2  (2  9.14) 2  (18  9.14) 2  192.86

s

c.

z

192.86  5.67 6

x  x 18  9.14   1.56 s 5.67

The 18 days required to restore service after hurricane Wilma is not an outlier. d.

34. a.

Yes, FP&L should consider ways to improve its emergency repair procedures. The mean, median and mode show repairs requiring an average of 8 to 9 days can be expected if similar hurricanes are encountered in the future. The 18 days required to restore service after hurricane Wilma should not be considered unusual if FP&L continues to use its current emergency repair procedures. With the number of customers affected running into the millions, plans to shorten the number of days to restore service should be undertaken by the company.

x

s

b.

z

xi 765   76.5 n 10

( xi  x )2 442.5  7 n 1 10  1 x  x 84  76.5   1.07 s 7

Approximately one standard deviation above the mean. Approximately 68% of the scores are within one standard deviation. Thus, half of (100-68), or 16%, of the games should have a winning score of 84 or more points.

z

x  x 90  76.5   1.93 s 7

Approximately two standard deviations above the mean. Approximately 95% of the scores are within two standard deviations. Thus, half of (100-95), or 2.5%, of the games should have a winning score of more than 90 points.

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Descriptive Statistics: Numerical Measures c.

x

s

xi 122   12.2 n 10

( xi  x )2 559.6   7.89 n 1 10  1

Largest margin 24: z 

35. a. b.

x

x

i

n



x  x 24  12.2   1.50 . No outliers. s 7.89

1050  75 14

$75.00 - $72.20 = $2.80 $2.80/$72.20 = .0388

c.

7th position – Green Bay Packers

63

8th position – Pittsburgh Steelers

67

Median = d.

Ticket price increased 3.88% during the one-year period.

63  67  65 2

i  .25(14)  3.5 Use 4th position Q1 = 61

(Tennessee Titans)

i  .75(14)  10.5 Use 11th position Q1 = 83

(Indianapolis Colts)

( xi  x )2 9504   27.04 n 1 13

e.

s

f.

Dallas Cowboys: z 

x





160  75  3.14 27.04

With z> 3, this is an outlier. The Dallas Cowboys have an unusually high ticket price compared to the other NFL teams.

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Chapter 3 36.

15, 20, 25, 25, 27, 28, 30, 34 Smallest = 15

i

25 (8)  2 100

Median 

i

Q1 

20  25  22.5 2

Q3 

28  30  29 2

25  27  26 2

75 (8)  8 100

Largest = 34 37.

38.

5, 6, 8, 10, 10, 12, 15, 16, 18 Smallest = 5

i

25 (9)  2.25 Q1 = 8 (3rd position) 100

Median = 10

i

75 (9)  6.75 Q3 = 15 (7th position) 100

Largest = 18

3 - 18 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part

Descriptive Statistics: Numerical Measures 39.

IQR = 50 - 42 = 8 Lower Limit:

Q1 - 1.5 IQR = 42 - 12 = 30

Upper Limit:

Q3 + 1.5 IQR = 50 + 12 = 62

65 is an outlier 40. a. b.

Subway with 29,612 locations. Ordered data: 1,397 2,558 2,615 4,516 5,110 5,619 5,889 8,053 8,082 10,238 11,553 24,799 29,612 Median n = 13; median is the middle value or 7th position Median = 5,889

c.

Smallest value = 1,397 i

25 25 ( n)  (13)  3.25 Round up to 4th position 100 100

Q1 = 4,516 i

75 75 ( n)  (13)  9.75 Round up to 10th position 100 100

Q3 = 10,238 Largest value = 29,612 5-number summary: 1,397 4,516 5,889 10,238 29,612 d.

Limits: IQR = Q3 - Q1 = 10,238 - 4,516 = 5,722 Lower Limit:

Q1 - 1.5 (IQR) = 4,516 – 1.5(5,722) = -4,067

Upper Limit:

Q3 + 1.5 (IQR) = 10,238 + 1.5(5,722) = 18,821

Both Subway and McDonald’s are outliers with a greater number of locations than other well-known food franchises. e.

0

5,000

10,000

15,000 3 - 19

20,000

*

*

25,000

30,000

© 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part. .

Chapter 3 41. a.

b.

The first place runner in the men’s group finished 109.03  65.30  43.73 minutes ahead of the first place runner in the women’s group. Lauren Wald would have finished in 11 th place for the combined groups.

 50  th th Men: i    22  .50(22)  11 . Use the 11 and 12 place finishes. 100   Median =

109.05  110.23  109.64 2

 50  th Women: i    31  .50(31)  15.5 . Use the 16 place finish. Median = 131.67.  100  Using the median finish times, the men’s group finished 131.67 109.64  22.03 minutes ahead of the women’s group. Also note that the fastest time for a woman runner, 109.03 minutes, is approximately equal to the median time of 109.64 minutes for the men’s group. c.

Men: Lowest time = 65.30; Highest time = 148.70 Q1:

 25  th i  n  .25(22)  5.5 Use 6 position. Q1 = 87.18  100 

Q3:

 75  th i  n  .75(22)  16.5 Use 17 position. Q3 = 128.40  100 

Five number summary for men: 65.30, 87.18, 109.64, 128.40, 148.70 Women: Lowest time = 109.03; Highest time = 189.28 Q1:

 25  th i  n  .25(31)  7.75 Use 8 position. Q1 = 122.08 100  

Q3:

 75  th i  n  .75(31)  23.25 Use 24 position. Q3 = 147.18  100 

Five number summary for women: 109.03, 122.08, 131.67, 147.18, 189.28 d.

Men: IQR = 128.40  87.18  41.22 Lower Limit = Q1  1.5( IQR)  87.18 1.5(41.22)  25.35 Upper Limit = Q3  1.5( IQR)  128.40  1.5(41.22)  190.23 There are no outliers in the men’s group. Women: IQR = 147.18 122.08  25.10 Lower Limit = Q1  1.5( IQR)  122.08 1.5(25.10)  84.43 3 - 20

© 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part

Descriptive Statistics: Numerical Measures

Upper Limit = Q3  1.5( IQR)  147.18  1.5(25.10)  184.83 The two slowest women runners with times of 189.27 and 189.28 minutes are outliers in the women’s group. e. Box Plot of Men and Women Runners 200

Time in Minutes

175 150

125 100

75 50 Men

Women

The box plots show the men runners with the faster or lower finish times. However, the box plots show the women runners with the lower variation in finish times. The interquartile ranges of 41.22 minutes for men and 25.10 minutes for women support this conclusion. 42. a.

Median n = 20; 10th and 11th positions Median =

b.

73  74  73.5 2

Smallest 68 Q1:

 25  th th i  20  5 ; 5 and 6 positions  100  Q1 

Q3:

71  72  71.5 2

 75  th th i  20  15 ; 15 and 16 positions  100  Q3 

74  75  74.5 2

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Chapter 3 Largest 77 5- number summary: 68, 71.5, 73.5, 74.5, 77 c.

IQR = Q3 – Q1 = 74.5 – 71.5 = 3 Lower Limit = Q1 – 1.5(IQR) = 71.5 – 1.5(3) = 67 Upper Limit = Q3 + 1.5(IQR) = 74.5 + 1.5(3) = 79 All ratings are between 67 and 79. There are no outliers for the T-Mobile service.

d.

Using the solution procedures shown in parts a, b, and c, the five number summaries and outlier limits for the other three cell-phone services are as follows. AT&T

66, 68, 71, 73, 75

Limits: 60.5 and 80.5

Sprint

63, 65, 66, 67.5, 69

Limits: 61.25 and 71.25

Verizon

75, 77, 78.5, 79.5, 81

Limits: 73.25 and 83.25

There are no outliers for any of the cell-phone services. e. Box Plots of Cell-Phone Services 80

Rating

75

70

65

AT&T

Sprint

T-Mobile

Verizon

The box plots show that Verizon is the best cell-phone service provider in terms of overall customer satisfaction. Verizon’s lowest rating is better than the highest AT&T and Sprint ratings and is better than 75% of the T-Mobile ratings. Sprint shows the lowest customer satisfaction ratings among the four services.

3 - 22 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part

Descriptive Statistics: Numerical Measures 43. a.

Total Salary for the Philadelphia Phillies = $96,870,000 Median n = 28; 14th and 15th positions Median =

900  1700  1300 2

Smallest 390  25  th th i Q1:  28  7 ; 7 and 8 positions 100  

Q1 

Q3:

425  440  432.5 2

 75  st nd i  28  21 ; 21 and 22 positions  100  Q3 

6000  6350  6175 2

Largest 14250 5- number summary for the Philadelphia Phillies: 390, 432.5, 1300, 6175, 14250 Using the 5-number summary, the lower quartile shows salaries closely bunched between 390 and 432.5. The median is 1300. The most variation is in the upper quartile where the salaries are spread between 6175 and 14250, or between $6,175,000 and $14,250,000. b.

IQR = Q3 – Q1 = 6175 – 432.5 = 5742.5 Lower Limit = Q1 – 1.5(IQR) = 432.5 –1.5(5742.5) = – 8181.25;

Use 0

Upper Limit = Q3 + 1.5(IQR) = 6175 + 1.5(5742.5) = 14788.75 All salaries are between 0 and 14788.75. There are no salary outliers for the Philadelphia Phillies. c.

Using the solution procedures shown in parts a and b, the total salary, the five-number summaries, and the outlier limits for the other teams are as follows. Los Angeles Dodgers $136,373,000 390, 403, 857.5, 9125, 19000

Limits: 0 and 22208

Tampa Bay Rays $ 42,334,000 390, 399, 415, 2350, 6000

Limits: 0 and 5276.5

Boston Red Sox $120,460,000 396, 439.5, 2500, 8166.5, 14000

Limits: 0 and 19757

3 - 23 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part. .

Chapter 3 The Los Angeles Dodgers had the highest payroll while the Tampa Bay Rays clearly had the lowest payroll among the four teams. With the lower salaries, the Rays had two outlier salaries compared to other salaries on the team. But these top two salaries are substantially below the top salaries for the other three teams. There are no outliers for the Phillies, Dodgers and Red Sox. d. Box Plots of Phillies, Dodgers, Rays, and Red Sox Salaries 20000

Salary ($1000's)

15000

10000

5000

0 Phillies

Dodgers

Rays

Red Sox

The box plots show that the lowest salaries for the four teams are very similar. The Red Sox have the highest median salary. Of the four teams the Dodgers have the highest upper end salaries and highest total payroll, while the Rays are clearly the lowest paid team. For this data, we would conclude that paying higher salaries do not always bring championships. In the National League Championship, the lower paid Phillies beat the higher paid Dodgers. In the American League Championship, the lower paid Rays beat the higher paid Red Sox. The biggest surprise was how the Tampa Bay Rays over achieved based on their salaries and made it to the World Series. Teams with the highest salaries do not always win the championships. 44. a.

x

xi 837.5   18.2 n 46

Median 23rd position 15.1 24th position 15.6 Median =

15.1  15.6  15.35 2

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Descriptive Statistics: Numerical Measures

b.

 25  i  n  .25(46)  11.5  100 

Q1:

12th position: Q1 = 11.7  75  i  n  .75(46)  34.5  100 

Q3:

35th position: Q3 = 23.5 c.

3.4, 11.7, 15.35, 23.5, 41.3

d.

IQR = 23.5 - 11.7 = 11.8 Lower Limit = Q1 - 1.5(IQR) = 11.7 - 1.5(11.8) = -6

Use 0

Upper Limit = Q3 + 1.5(IQR) = 23.5 + 1.5(11.8) = 41.2 Yes, one: Alger Small Cap 41.3

* 0

10

20

30

40

45. a.

70 60 50

y

40 30 20 10 0 0

5

10

15

20

x 3 - 25 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part. .

Chapter 3 b.

Negative relationship

c/d. xi  40

x

40 8 5

yi  230

( xi  x )( yi  y )  240

y

( xi  x ) 2  118

sxy 

( xi  x )( yi  y ) 240   60 n 1 5 1

sx 

( xi  x ) 2 118   5.4314 n 1 5 1

sy 

( yi  y ) 2  n 1

rxy 

sxy sx s y



230  46 5 ( yi  y ) 2  520

520  11.4018 5 1

60  .969 (5.4314)(11.4018)

There is a strong negative linear relationship. 46. a.

18 16 14 12 y

10 8 6 4 2 0 0

5

10

15

20

25

30

x

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Descriptive Statistics: Numerical Measures b.

Positive relationship 80  16 c/d. xi  80 x  5

( xi  x )( yi  y )  106

yi  50

y

50  10 5

( xi  x ) 2  272

sxy 

( xi  x )( yi  y ) 106   26.5 n 1 5 1

sx 

( xi  x )2 272   8.2462 n 1 5 1

( yi  y ) 2  86

( yi  y )2 86   4.6368 n 1 5 1 sxy 26.5 rxy    .693 sx s y (8.2462)(4.6368) sy 

A positive linear relationship 47. a.

b.

The scatter diagram shows a positive relationship with higher predicted point margins associated with higher actual point margins.

c.

Let x = predicted point margin and y = actual point margin xi  30

x

30 3 10

( xi  x )( yi  y )  201

sxy 

yi  110

y

110  11 10

( xi  x )2  276

( yi  y )2  458

( xi  x )( yi  y ) 201   22.3333 n 1 10  1

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Chapter 3

A positive covariance shows a positive relationship between predicted point margins and actual point margins. d.

sx 

( xi  x ) 2 276   5.5377 n 1 10  1

sy 

( yi  y ) 2 458   7.1336 n 1 10  1

rxy 

sxy sx s y



22.3333  .565 (5.5377)(7.1336)

The modest positive correlation shows that the Las Vegas predicted point margin is a general, but not a perfect, indicator of the actual point margin in college football bowl games. Note: The Las Vegas odds makers set the point margins so that someone betting on a favored team has to have the team win by more than the point margin to win the bet. For example, someone betting on Auburn to win the Outback Bowl would have to have Auburn win by more than five points to win the bet. Since Auburn beat Northwestern by only three points, the person betting on Auburn would have lost the bet. A review of the predicted and actual point margins shows that the favorites won by more than the predicted point margin in five bowl games: Gator, Sugar, Cotton, Alamo, and the Championship bowl game. The underdog either won its game or kept the actual point margin less than the predicted point margin in the other five bowl games. In this case, betting on the underdog would have provided winners in the Outback, Capital One, Rose, Fiesta and Orange bowls. In this example, the Las Vegas odds point margins made betting on the favored team a 50-50 probability of winning the bet.

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Descriptive Statistics: Numerical Measures 48.

Let x = miles per hour and y = miles per gallon xi  420

x

420  42 10

( xi  x )( yi  y )  475

sxy 

yi  270

( xi  x )2  1660

270  27 10

( yi  y )2  164

( xi  x )( yi  y ) 475   52.7778 n 1 10  1

sx 

( xi  x ) 2 1660   13.5810 n 1 10  1

sy 

( yi  y ) 2 164   4.2687 n 1 10  1

rxy 

y

sxy sx s y



52.7778  .91 (13.5810)(4.2687)

A strong negative linear relationship exists. For driving speeds between 25 and 60 miles per hour, higher speeds are associated with lower miles per gallon.

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Chapter 3

49. a.

x

xi 184   6.81 n 27

xi

yi

7.1 5.2 7.8 7.8 5.8 5.8 9.3 5.7 7.3 7.6 8.2 7.1 6.3 6.6 6.2 6.3 7.0 6.2 5.5 6.5 6.0 8.3 7.5 7.1 6.8 5.5 7.5

sxy 

7.02 5.31 5.38 5.40 5.00 4.07 6.53 5.57 6.99 11.12 7.56 12.11 4.39 4.78 5.78 6.08 10.05 4.75 7.22 3.79 3.62 9.24 4.40 6.91 5.57 3.87 8.42

y

yi 170.93   6.33 n 27

( xi  x )

( yi  y )

( xi  x )2

( yi  y )2

( xi  x )( yi  y )

0.2852 -1.6148 0.9852 0.9852 -1.0148 -1.0148 2.4852 -1.1148 0.4852 0.7852 1.3852 0.2852 -0.5148 -0.2148 -0.6148 -0.5148 0.1852 -0.6148 -1.3148 -0.3148 -0.8148 1.4852 0.6852 0.2852 -0.0148 -1.3148 0.6852

0.6893 -1.0207 -0.9507 -0.9307 -1.3307 -2.2607 0.1993 -0.7607 0.6593 4.7893 1.2293 5.7793 -1.9407 -1.5507 -0.5507 -0.2507 3.7193 -1.5807 0.8893 -2.5407 -2.7107 2.9093 -1.9307 0.5793 -0.7607 -2.4607 2.0893 Total

0.0813 2.6076 0.9706 0.9706 1.0298 1.0298 6.1761 1.2428 0.2354 0.6165 1.9187 0.0813 0.2650 0.0461 0.3780 0.2650 0.0343 0.3780 1.7287 0.0991 0.6639 2.2058 0.4695 0.0813 0.0002 1.7287 0.4695 25.77407

0.4751 1.0419 0.9039 0.8663 1.7709 5.1109 0.0397 0.5787 0.4346 22.9370 1.5111 33.3998 3.7665 2.4048 0.3033 0.0629 13.8329 2.4987 0.7908 6.4554 7.3481 8.4638 3.7278 0.3355 0.5787 6.0552 4.3650 130.0594

0.1966 1.6483 -0.9367 -0.9170 1.3505 2.2942 0.4952 0.8481 0.3199 3.7605 1.7028 1.6482 0.9991 0.3331 0.3386 0.1291 0.6888 0.9719 -1.1692 0.7999 2.2088 4.3208 -1.3229 0.1652 0.0113 3.2354 1.4315 25.5517

( xi  x )( yi  y ) 25.5517   .9828 n 1 26

sx 

( xi  x ) 2  n 1

sy 

( yi  y ) 2 130.0594   2.2366 n 1 26

rxy 

sxy sx s y



25.7741  .9956 26

.9828  .44 (.9956)(2.2366)

There is evidence of a modest positive linear association between the jobless rate and the delinquent housing loan percentage. If the jobless rate were to increase, it is likely that an increase in the percentage of delinquent housing loans would also occur.

3 - 30 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part

Descriptive Statistics: Numerical Measures b.

50. a.

1

S&P 500

0.5

0 -1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

-0.5

-1 DJIA

b.

x

xi 1.44   .16 n 9

y

xi 1.17   .13 n 9

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Chapter 3

xi

yi

( xi  x )

0.20 0.82 -0.99 0.04 -0.24 1.01 0.30 0.55 -0.25

0.24 0.19 -0.91 0.08 -0.33 0.87 0.36 0.83 -0.16

0.04 0.66 -1.15 -0.12 -0.40 0.85 0.14 0.39 -0.41

sxy 

( xi  x )2 0.0016 0.4356 1.3225 0.0144 0.1600 0.7225 0.0196 0.1521 0.1681 2.9964

0.11 0.06 -1.04 -0.05 -0.46 0.74 0.23 0.70 -0.29 Total

( yi  y )2 0.0121 0.0036 1.0816 0.0025 0.2166 0.5476 0.0529 0.4900 0.0841 2.4860

( xi  x )( yi  y ) 0.0044 0.0396 1.1960 0.0060 0.1840 0.6290 0.0322 0.2730 0.1189 2.4831

( xi  x )( yi  y ) 2.4831   .3104 n 1 8

sx 

( xi  x ) 2  n 1

2.9964  .6120 8

sy 

( yi  y ) 2  n 1

2.4860  .5574 8

rxy  c.

( yi  y )

sxy sx s y



.3104  .91 (.6120)(.5574)

There is a strong positive linear association between DJIA and S&P 500. If you know the change in either, you will have a good idea of the stock market performance for the day.

51. a.

x

xi 945   67.5 n 14

b.

y

yi 706   50.4286  50.4 n 14

3 - 32 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part

Descriptive Statistics: Numerical Measures c.

sxy 

xi

yi

( xi  x )

( yi  y )

( xi  x )2

( yi  y )2

( xi  x )( yi  y )

68 70 65 96 57 70 80 67 44 69 76 69 70 44

50 49 44 64 46 45 73 45 29 44 69 51 58 39

.5 2.5 -2.5 28.5 -10.5 2.5 12.5 -.5 -23.5 1.5 8.5 1.5 2.5 -23.5

-.4286 -1.4286 -6.4286 13.5714 -4.4286 -5.4286 22.5714 -5.4286 -21.4286 -6.4286 18.5714 .5714 7.5714 -11.4286 Total

.25 6.25 6.25 812.25 110.25 6.25 156.25 .25 552.25 2.25 72.25 2.25 6.25 552.25 2285.5

.1837 2.0408 41.3265 184.1837 19.6122 29.4694 509.4694 29.4694 459.1837 41.3265 344.8980 .3265 57.3265 130.6122 1849.4286

-.2143 -3.5714 16.0714 386.7857 46.5000 -13.5714 282.1429 2.7143 503.5714 -9.6429 157.8571 .8571 18.9286 268.5714 1657.0000

( xi  x )( yi  y ) 1657   127.4615 n 1 14  1

sx 

( xi  x ) 2  n 1

sy 

( yi  y )2 1849.4286   11.9274 n 1 14  1

rxy 

sxy sx s y



2285.5  13.2592 14  1

127.4615  .81 13.2592(11.9274)

High positive correlation as should be expected. 52. a.

b.

x

wi xi 6(3.2)  3(2)  2(2.5)  8(5) 70.2    3.69 wi 6 3 2 8 19

3.2  2  2.5  5 12.7   3175 . 4 4

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Chapter 3 53. fi 4 7 9 5 25

x

s2 

Mi 5 10 15 20

fi Mi 20 70 135 100 325

f i M i 325   13 n 25 fi

Mi

Mi  x

(M i  x )2

fi ( M i  x )2

4 7 9 5

5 10 15 20

-8 -3 +2 +7

64 9 4 49

256 63 36 245 600

fi ( M i  x )2 600   25 n 1 24

s  25  5 54. a. Grade xi 4 (A) 3 (B) 2 (C) 1 (D) 0 (F)

x b. 55. a.

Weight Wi 9 15 33 3 0 60 Credit Hours

wi xi 9(4)  15(3)  33(2)  3(1) 150    2.50 wi 9  15  33  3 60

Yes; satisfies the 2.5 grade point average requirement x

fi M i 9191(4.65)  2621(18.15)  1419(11.36)  2900(6.75)  N 9191  2621  1419  2900 126, 004.14   7.81 16,131

The weighted average total return for the Morningstar funds is 7.81%. b.

If the amount invested in each fund was available, it would be better to use those amounts as weights. The weighted return computed in part (a) will be a good approximation, if the amount invested in the various funds is approximately equal.

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Descriptive Statistics: Numerical Measures c.

2000(4.65)  4000(18.15)  3000(11.36)  1000(6.75) 2000  4000  3000  1000 122, 730   12.27 10, 000

Portfolio Return =

The portfolio return would be 12.27%. 56. Assessment 5 4 3 2 1 Total

Deans 44 66 60 10 0 180

x

fi M i 684   3.8 n 180

Recruiters: x 

fi M i 444   3.7 n 120

Deans:

fiMi 220 264 180 20 0 684

Recruiters 31 34 43 12 0 120

fiMi 155 136 129 24 0 444

57. a. Price per Share $0-9 $10-19 $20-29 $30-39 $40-49 $50-59 $60-69 $70-79 $80-89 $90-99 Total x

Frequency 4 5 7 3 4 4 0 2 0 1 30

Midpoint 4.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5 94.5

fiMi 18.0 72.5 171.5 103.5 178.0 218.0 0.0 149.0 0.0 94.5 1005.0

Mi  x

i fM i 1005.0   33.50 n 30

Price per Share

Frequency

Midpoint

$0-9 $10-19 $20-29 $30-39 $40-49 $50-59 $60-69 $70-79 $80-89 $90-99

4 5 7 3 4 4 0 2 0 1

4.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5 94.5

-29 -19 -9 1 11 21 31 41 51 61

( Mi  x ) 2

841 361 81 1 121 441 961 1681 2601 3721 Total

f i ( Mi  x ) 2

3364 1805 567 3 484 1764 0 3362 0 3721 15070

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Chapter 3

fi ( M i  x )2 15070   22.80 n 1 29

s b.

58. a.

The mean price per share had decreased ($45.83-$33.50)=$12.33, or (12.33/45.83)(100) = 26.9% over the three-year period. The standard deviation had increased from $18.14 to $22.80 over the same three-year period. In January 2009, the stock market as measured by the Dow Jones Industrial Average companies had declined and was showing more variability.

xi 27000   1800 n 15

x

Median 8th position = 1351 b.

 25  i 15  3.75  100 

Q1:

4th position: Q1 = 387  75  i 15  11.25  100 

Q3:

c.

12th position: Q3 = 1710 Range = 7450 - 170 = 7280 IQR = Q3 - Q1 = 1710 - 387 = 1323

d.

s2 

( xi  x )2 51, 454, 242   3, 675,303 n 1 15  1

s  3,675,303  1917 e.

High positive skewness. This seems reasonable. A relatively few people will have large monthly expenditures causing the right tail of the distribution to become longer.

f.

z

x  x 4135  1800   1.22 s 1917

z

x  x 7450  1800   2.95 do not indicate outliers. s 1917

These values of z do not indicate outliers. However, the upper limit for outliers is Q3 + 1.5(IQR) = 1710 + 1.5(1323) = 3695 Thus, both $4135 and $7450 are outliers.

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Descriptive Statistics: Numerical Measures 59. a.

Arrange the data in order Men 21 23 24 25 25 26 26 27 27 27 27 28 28 29 30 30 32 35 Median i = .5(18) = 9 Use 9th and 10th positions Median = 27 Women 19 20 22 22 23 23 24 25 25 26 26 27 28 29 30 Median i = .5(15) = 7.5 Use 8th position Median = 25

b. Q1

Q3

c.

60. a.

Men i = .25(18) = 4.5 Use 5th position Q1 = 25

Women i = .25(15) = 3.75 Use 4th position Q1 = 22

i = .75(18) = 13.5 Use 14th position Q3 = 29

i = .75(15) = 11.25 Use 12th position Q3 = 27

Young people today are waiting longer to get married than young people did 25 years ago. The median age for men has increased from 25 to 27. The median age for women has increased from 22 to 25.

x

xi 100   $10, 000 n 10

Mean debt upon graduation is $10,000. b.

s2 

( xi  x )2 221.78   24.64 n 1 9

s  24.64  4.96 61. a.

x

xi 23   2.3 n 10

Median: 5th and 6th positions Median =

1.8  1.9  1.85 2 3 - 37

© 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part. .

Chapter 3

b.

s2 

( xi  x )2 17.08   1.90 n 1 10  1

s  1.90  1.38 c.

Altria Group at 5%

d.

z

x  x 1.6  2.3   .51 s 1.38

McDonald's is about 1/2 a standard deviation below the mean dividend yield. e.

z

x  x 3.7  2.3   1.02 s 1.38

General Motors is about one standard deviation above the mean dividend yield. f.

Altria Group

z

x  x 5.0  2.3   1.96 s 1.38

Wal-Mart

z

x  x 0.7  2.3   1.16 s 1.38

No outliers.

62. a.

x

b.

s

c.

z

xi 13, 400   670 n 20

( xi  x )2 3,949, 200   $456 n 1 20  1 x  x 2040  670   3.00 s 456

Yes it is an outlier. d.

63. a.

First of all, the employee payroll service will be up to date on tax regulations. This will save the small business owner the time and effort of learning tax regulations. This will enable the owner greater time to devote to other aspects of the business. In addition, a correctly filed employment tax return will reduce the potential of a tax penalty. Public Transportation: x  Automobile: x 

320  32 10

320  32 10

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Descriptive Statistics: Numerical Measures b.

Public Transportation: s = 4.64 Automobile: s = 1.83

c.

Prefer the automobile. The mean times are the same, but the auto has less variability.

d.

Data in ascending order: Public:

25 28 29 29 32 32 33 34 37 41

Auto:

29 30 31 31 32 32 33 33 34 35

Five number Summaries Public:

25 29 32 34 41

Auto:

29 31 32 33 35

Box Plots: Public:

Auto:

The box plots do show lower variability with automobile transportation and support the conclusion in part c. 64. a.

Arrange the data in ascending order 48.8 92.6 111.0 …….. 958.0 995.9 2325.0 With n = 14, the median is the average of home prices in position 7 and 8. Median home price =

212.9  218.9  215.9 2

Median home price = $215,900

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Chapter 3

b.

215,900  139,300  .55 139,300 55% increase over the five-year period

c.

n = 14

i

25 (n)  3.5 100

Use the 4th position Q1 = 175.0

i

75 (n)  10.5 100

Use the 11th position Q3 = 628.3

Q3  d.

362.5  628.3  495.4 2

Lowest price = 48.8 and highest price = 2324.0. Five-number summary: 48.8, 175.0, 215.9, 628.3, 2325.0

e.

IQR = Q3 - Q1 = 628.3 – 175.0 = 453.3 Upper limit = Q3 + 1.5IQR = 628.3 + 1.5(679.95) = 1308.25 Any price over $1,308,250 is an outlier. Yes, the price $2,325,000 is an outlier.

f.

x

xi 6749.4   482.1 n 14

The mean is sensitive to extremely high home prices and tends to overstate the more typical midrange home price. The sample mean of $482,100 has 79% of home prices below this value and 21% of the home prices above this value while the sample median $215,900 has 50% above and 50% below. The median is more stable and not influenced by the extremely high home prices. Using the sample mean $482,100 would overstate the more typical or middle home price.

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Descriptive Statistics: Numerical Measures 65. a.

Median for n = 50; Use 25th and 26th positions 25th – South Dakota 16.8 26th – Pennsylvania 16.9 Median =

b.

16.8  16.9  16.85% 2

 25  i  50  12.5  100 

Q1:

13th position: Q1 = 13.7% (Iowa)  75  i  50  37.5  100 

Q3:

38th position: Q3 = 20.2% (North Carolina & Georgia) 25% of the states have a poverty level less than or equal to 13.7% and 25% of the states have a poverty level greater than or equal to 20.2% c.

IQR = Q3 - Q1 = 20.2 – 13.7 = 6.5 Upper Limit = Q3 + 1.5(IQR) = 20.2 + 1.5(6.5) = 29.95 Lower Limit = Q1 - 1.5(IQR) = 13.7 - 1.5(6.5) =

3.95

Box Plot of Poverty % 30

Poverty %

25

20

15

10

The Minitab box plot shows the distribution of poverty levels is skewed to the right (positive). There are no states considered outliers. Mississippi with 29.5% is closest to being an outlier on the high poverty rate side. New Hampshire has the lowest poverty level with 9.6%. The five-number summary is 9.6, 13.7, 16.85, 20.2 and 29.95.

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Chapter 3

d.

The states in the lower quartile are the states with the lowest percentage of children who have lived below the poverty level in the last 12 months. These states are as follows. State

Region

Poverty %

New Hampshire

NE

9.6

Maryland

NE

9.7

Connecticut

NE

11.0

Hawaii

W

11.4

New Jersey

NE

11.8

Utah

W

11.9

Wyoming

W

12.0

Minnesota

MW

12.2

Virginia

SE

12.2

Massachusetts

NE

12.4

North Dakota

MW

13.0

Vermont

NE

13.2

Generally, these states are the states with better economic conditions and less poverty. The Northeast region with 6 of the 12 states in this quartile appears to be the best economic region of the country. The West region was second with 3 of the 12 states in this group. 66. a.

x

xi 4368   364 rooms n 12

b.

y

yi 5484   $457 n 12

c.

It is difficult to see much of a relationship. When the number of rooms becomes larger, there is no indication that the cost per night increases. The cost per night may even decrease slightly.

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Descriptive Statistics: Numerical Measures d.

xi

yi

220 727 285 273 145 213 398 343 250 414 400 700

499 340 585 495 495 279 279 455 595 367 675 420

sxy 

( xi  x )

( yi  y )

-144 363 -79 -91 -219 -151 34 -21 -114 50 36 336

42 -117 128 38 38 -178 -178 -2 138 -90 218 -37 Total

( xi  x )2

20.736 131,769 6,241 8,281 47,961 22,801 1,156 441 12,996 2,500 1,296 112,896 69,074

1,764 13,689 16,384 1,444 1,444 31,684 31,684 4 19,044 8,100 47,524 1,369 174,134

( xi  x )( yi  y ) -6,048 -42,471 -10,112 -3,458 -8,322 26,878 -6,052 42 -15,732 -4,500 7,848 -12,432 -74,359

( xi  x )( yi  y ) 74,350   6759.91 n 1 11

sx 

( xi  x ) 2 369, 074   183.17 n 1 11

sy 

( yi  y ) 2 174,134   125.82 n 1 11

rxy 

( yi  y )2

sxy sx s y



6759.91  .29 (183.17)(125.82)

There is evidence of a slightly negative linear association between the number of rooms and the cost per night for a double room. Although this is not a strong relationship, it suggests that the higher room rates tend to be associated with the smaller hotels. This tends to make sense when you think about the economies of scale for the larger hotels. Many of the amenities in terms of pools, equipment, spas, restaurants, and so on exist for all hotels in the Travel + Leisure top 50 hotels in the world. The smaller hotels tend to charge more for the rooms. The larger hotels can spread their fixed costs over many room and may actually be able to charge less per night and still achieve and nice profit. The larger hotels may also charge slightly less in an effort to obtain a higher occupancy rate. In any case, it appears that there is a slightly negative linear association between the number of rooms and the cost per night for a double room at the top hotels.

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Chapter 3 67. a. The scatter diagram is shown below.

The sample correlation coefficient is .954. This indicates a strong positive linear relationship between Morningstar’s Fair Value estimate per share and the most recent price per share for the stock.

b.

The scatter diagram is shown below:

The sample correlation coefficient is .624. While not a strong of a relationship as shown in part a, this indicates a positive linear relationship between Morningstar’s Fair Value estimate per share and the earnings per share for the stock.

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Descriptive Statistics: Numerical Measures 68. a. xi

yi

.407 .429 .417 .569 .569 .533 .724 .500 .577 .692 .500 .731 .643 .448

.422 .586 .546 .500 .457 .463 .617 .540 .549 .466 .377 .599 .488 .531

sxy 

( yi  y )

( xi  x )2

( yi  y )2

( xi  x )( yi  y )

-.1458 -.1238 -.1358 .0162 .0162 -.0198 .1712 -.0528 .0242 .1392 -.0528 .1782 .0902 -.1048

-.0881 .0759 .0359 -.0101 -.0531 -.0471 .1069 .0299 .0389 -.0441 -.1331 .0889 -.0221 .0209 Total

.0213 .0153 .0184 .0003 .0003 .0004 .0293 .0028 .0006 .0194 .0028 .0318 .0081 .0110 .1617

.0078 .0058 .0013 .0001 .0028 .0022 .0114 .0009 .0015 .0019 .0177 .0079 .0005 .0004 .0623

.0128 -.0094 -.0049 -.0002 -.0009 .0009 .0183 -.0016 .0009 -.0061 .0070 .0158 -.0020 -.0022 .0287

( xi  x )( yi  y ) .0287   .0022 n 1 14  1

sx 

( xi  x )2 .1617   .1115 n 1 14  1

sy 

( yi  y )2 .0623   .0692 n 1 14  1

rxy 

b.

( xi  x )

sxy sx s y



.0022  .286 .1115(.0692)

There is a low positive correlation between a major league baseball team’s winning percentage during spring training and its winning percentage during the regular season. The spring training record should not be expected to be a good indicator of how a team will play during the regular season. Spring training consists of practice games between teams with the outcome as to who wins or who loses not counting in the regular season standings or affecting the chances of making the playoffs. Teams use spring training to help players regain their timing and evaluate new players. Substitutions are frequent with the regular or better players rarely playing an entire spring training game. Winning is not the primary goal in spring training games. A low correlation between spring training winning percentage and regular season winning percentage should be anticipated.

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Chapter 3

69. fi

Mi

fi Mi

Mi  x

10 40 150 175 75 15 10 475

47 52 57 62 67 72 77

470 2080 8550 10850 5025 1080 770 28,825

-13.68 -8.68 -3.68 +1.32 +6.32 +11.32 +16.32

a.

x

28,825  60.68 475

b.

s2 

14,802.64  31.23 474

( Mi  x ) 2 187.1424 75.3424 13..5424 1.7424 39.9424 128.1424 266.3424

f i ( Mi  x ) 2 1871.42 3013.70 2031.36 304.92 2995.68 1922.14 2663.42 14,802.64

s  31.23  5.59

70.

x

wi xi 20(20)  30(12)  10(7)  15(5)  10(6) 965    11.4 days wi 20  30  10  15  10 85

3 - 46 © 2011 Cengage Learning. All Rights Reserved. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. May not be scanned, copied, duplicated, or posted to a publicly accessible website, in whole or in part