Statistics for Business and Economics chapter 03

Chapter Descriptive Statistics: Numerical Measures Learning Objectives Understand the purpose of measures of location Be able to compute the mean, median, mode, quartiles, and various percentiles Understand the purpose of measures of variability Be able to compute the range, interquartile range, variance, standard deviation, and coefficient of variation 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 Understand how z scores are computed and how they are used as a measure of relative location of a data value 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 Learn how to construct a 5-number summary and a box plot Be able to compute and interpret covariance and correlation as measures of association between two variables 10 Be able to compute a weighted mean Solutions: 3-1 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter x= Σxi 75 = = 15 n 10, 12, 16, 17, 20 Median = 16 (middle value) x= Σxi 96 = = 16 n 10, 12, 16, 17, 20, 21 Median = 15, 20, 25, 25, 27, 28, 30, 34 i= 20 (8) = 1.6 100 2nd position = 20 i= 25 (8) = 100 20 + 25 = 22.5 i= 65 (8) = 5.2 100 6th position = 28 i= 75 (8) = 100 28 + 30 = 29 Mean = 16 + 17 = 16.5 Σxi 657 = = 59.73 n 11 Median = 57 6th item Mode = 53 It appears times Σxi 3181 = = $159 n 20 a x= b Median 10th $160 11th $162 Median = Los Angeles Seattle 160 + 162 = $161 c Mode = $167 San Francisco and New Orleans d  25  i=  20 =  100  5th $134 3-2 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures 6th Q1 = e $139 134 + 139 = $136.50  75  i=  20 = 15  100  15th $167 16th $173 Q3 = 167 + 173 = $170 a x= Σxi 350 = = 18.42 n 19 b x= Σxi 120 = = 6.32 n 19 c 120 (100) = 34.3% of 3-point shots were made from the 20 feet, inch line during the 19 games 350 d Moving the 3-point line back to 20 feet, 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, 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, inch 3point line Σxi 148 = = 14.8 n 10 a x= b Order the data from low 6.7 to high 36.6 Median  50  i= ÷10 = Use 5th and 6th positions  100  Median = 10.1 + 16.1 = 13.1 c Mode = 7.2 (occurs times) d  25  i= ÷10 = 2.5 Use 3rd position Q1 = 7.2  100  3-3 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter  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 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  i= ÷20 = 10  100  Use 10th and 11 th positions  130 + 140  Median =  ÷ = 135   Mode = 120 (occurs times) b  25  th th i= ÷20 = Use and positions  100   115 + 115  Q1 =  ÷ = 115    75  i= ÷20 = 15  100  Use 15 th and 16 th positions  180 + 195  Q3 =  ÷ = 187.5   c  90  i= ÷20 = 18  100  Use 18th and 19th positions  235 + 255  90th percentile =  ÷ = 245   90% of the tax returns cost $245 or less 10% of the tax returns cost $245 or more a Ordered data: 112.8 140.2 169.9 177.5 3-4 181.3 202.5 230.0 315.5 470.2 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures 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 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 10 a b Minimum = 4%; Maximum = 3.5% Σxi = 69 x= Σxi 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 economists c For Q1,  25  th i= ÷30 = 7.5; round up and use the item  100  Q1 = 2.0% For Q3,  75  i= ÷30 = 22.5; round up and use the 23rd item  100  3-5 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 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 12 Disney Total Revenue: $3,321 million (13 movies) x= Σxi 3321 = = $255.5 n 13 104 110 136 169 249 250 253 273 304 325 346 354 448 Median 7th position Median = $253 Q1: i = 25(13) = 3.25 4th position Q1 = $169 Q3: i = 75(13) = 9.75 10th position Q3 = $325 Pixar Total Revenue: $3,231 million (6 movies) x= Σxi 3231 = = $538.5 n 362 363 485 525 631 865 3-6 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures Median (3rd and 4th positions) Median = 485 + 525 = $505 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 films Mean Median Disney $225.5 $253 Pixar $538.5 $505 The first quartiles show 75% of Disney films better than $169 million while 75% of Pixar films better than $363 million The third quartiles show 25% of Disney films better than $325 million while 25% of Pixar films 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 = 14 x= Σxi 75 = = 15 n s2 = Σ( xi − x ) 64 = = 16 n −1 s = 16 = 15 15, 20, 25, 25, 27, 28, 30, 34 Range = 34 - 15 = 19 3-7 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter i= 25 (8) = 100 Q1 = 20 + 25 = 22.5 i= 75 (8) = 100 Q3 = 28 + 30 = 29 IQR = Q3 - Q1 = 29 - 22.5 = 6.5 x= Σxi 204 = = 255 n s2 = Σ( xi − x ) 242 = = 34.57 n −1 s = 34.57 = 588 16 a b Range = 190 - 168 = 22 Σ( xi − x ) = 376 s = 376 = 75.2 c s = 75.2 = 8.67 d  8.67  Coefficient of Variation =  100% = 4.87%  178  17 a b With DVD x= Σxi 2050 = = 410 n Without DVD x= Σxi 1550 = = 310 n With DVD $410 - $310 = $100 more expensive With DVD Range = 500 - 300 = 200 s2 = Σ( xi − x )2 22000 = = 5500 n −1 s = 5500 = 74.2 Without DVD Range = 360 - 290 = 70 s2 = Σ( xi − x )2 3200 = = 800 n −1 s = 800 = 28.3 3-8 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures 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= Σxi 266 = = $38 per day n s2 = Σ( xi − x )2 582 = = 97 n −1 s = 97 = $9.85 b 19 a 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 Range = 60 - 28 = 32 IQR = Q3 - Q1 = 55 - 45 = 10 b x= 435 = 48.33 Σ( xi − x ) = 742 s2 = Σ( xi − x )2 742 = = 92.75 n −1 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 - = 4.1 = 0.67 J.C Clark: Range = 15 - = s= s= 21 a 60.1 = 2.58 Cities: x= Σxi 198 = = $33 n s2 = Σ( xi − x )2 72 = = 14.40 n −1 −1 s = 14.40 = 3.79 3-9 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter Retirement Areas: x= Σxi 192 = = $32 n s2 = Σ( xi − x )2 18 = = 3.60 n −1 −1 s = 3.60 = 1.90 b 22 a 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 Freshmen x = Seniors x= Σxi 32125 = = 1285 n 25 Σxi 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 =  100  Q1 = 368 + 373 = 370.5  75  i= ÷20 = 15  100  Q1 = 489 + 515 = 502 IQR = Q3 - Q1 = 502 – 370.5 = 131.5 d s= Σ( xi − x ) n −1 Freshmen s = 3233186 = 367.04 24 - 10 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures b x= Σxi 1.44 = = 16 n y= Σxi 1.17 = = 13 n 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 ) 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 ) 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 Σ( xi − x )( yi − y ) 2.4831 = = 3104 n −1 sx = Σ( xi − x ) = n −1 2.9964 = 6120 sy = Σ( yi − y ) = n −1 2.4860 = 5574 rxy = c ( yi − y ) s xy sx s y = 3104 = 910 (.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 - 29 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 51 a x= Σxi 945 = = 67.5 n 14 b y= Σyi 706 = = 50.4286 ≈ 50.4 n 14 c sxy = xi yi ( xi − x ) ( yi − y ) ( xi − x ) ( yi − y ) ( 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 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 Σ( xi − x )( yi − y ) 1657 = = 127.4615 n −1 14 − sx = Σ( xi − x ) = n −1 sy = Σ( yi − y ) 1849.4286 = = 11.9274 n −1 14 − rxy = sxy sx s y = 2285.5 = 13.2592 14 − 127.4615 = +.806 13.2592(11.9274) High positive correlation as should be expected 52 a b x= Σwi xi 6(3.2) + 3(2) + 2(2.5) + 8(5) 70.2 = = = 3.69 Σwi 6+ 3+ +8 19 3.2 + + 2.5 + 12.7 = = 3175 4 53 fi Mi - 30 fi M i 20 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures 25 x= s2 = 10 15 20 70 135 100 325 Σf i M i 325 = = 13 n 25 fi Mi Mi − x 5 10 15 20 -8 -3 +2 +7 (M i − x )2 64 49 fi ( M i − x )2 256 63 36 245 600 Σf i ( M i − x ) 600 = = 25 n −1 24 s = 25 = 54 a Grade xi (A) (B) (C) (D) (F) x= b 55 a Weight Wi 15 33 60 Credit Hours Σwi xi 9(4) + 15(3) + 33(2) + 3(1) 150 = = = 2.50 Σwi + 15 + 33 + 60 Yes; satisfies the 2.5 grade point average requirement x= Σf i 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 c Portfolio Return = 2000(4.65) + 4000(18.15) + 3000(11.36) + 1000(6.75) 2000 + 4000 + 3000 + 1000 122, 730 = = 12.27 10, 000 - 31 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter The portfolio return would be 12.27% 56 Assessment Total Deans 44 66 60 10 180 x= Σf i M i 684 = = 3.8 n 180 Recruiters: x = Σf i M i 444 = = 3.7 n 120 Deans: f i Mi 220 264 180 20 684 Recruiters 31 34 43 12 120 f i Mi 155 136 129 24 444 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= Midpoint 4.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5 94.5 f i Mi 18.0 72.5 171.5 103.5 178.0 218.0 0.0 149.0 0.0 94.5 1005.0 Mi − x Σi 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 4.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5 94.5 s= b Frequency 4 30 -29 -19 -9 11 21 31 41 51 61 ( Mi − x )2 841 361 81 121 441 961 1681 2601 3721 Total f i ( Mi − x )2 3364 1805 567 484 1764 3362 3721 15070 Σf i ( M i − x ) 15070 = = 22.80 n −1 29 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 - 32 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures 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 58 a x= Σxi 27000 = = 1800 n 15 Median 8th position = 1351 b Q1: Q3:  25  i= 15 = 3.75  100  4th position: Q1 = 387  75  i= 15 = 11.25  100  12th position: Q3 = 1710 c Range = 7450 - 170 = 7280 IQR = Q3 - Q1 = 1710 - 387 = 1323 d s2 = Σ( xi − x )2 51, 454, 242 = = 3, 675,303 n −1 15 − 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 not indicate outliers s 1917 These values of z 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 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) = - 33 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 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= Σxi 23 = = 2.3 n 10 Median: 5th and 6th positions Median = b s2 = 1.8 + 1.9 = 1.85 Σ( xi − x )2 17.08 = = 1.90 n −1 10 − 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 - 34 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures 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 61 a Σxi 100 = = $10, 000 n 10 x= Mean debt upon graduation is $10,000 b s2 = Σ( xi − x )2 221.78 = = 24.64 n −1 s = 24.64 = 4.96 62 a x= b s= c z= Σxi 13, 400 = = 670 n 20 Σ( xi − x ) = n −1 3,949, 200 = $456 20 − 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 = b 320 = 32 10 320 = 32 10 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 - 35 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter Five number Summaries Public: 25 29 32 34 41 Auto: 29 31 32 33 35 Box Plots: Public: 24 28 32 36 40 28 32 36 40 Auto: 24 The box plots 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 and Median home price = 212.9 + 218.9 = 215.9 Median home price = $215,900 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 - 36 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures i= 75 (n) = 10.5 100 Use the 11th position Q3 = 628.3 Q3 = d 362.5 + 628.3 = 495.4 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= Σxi 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 65 a Median for n = 50; Use 25th and 26th positions 25th – South Dakota 16.8 26th – Pennsylvania 16.9 Median = b Q1: 16.8 + 16.9 = 16.85%  25  i= ÷50 = 12.5  100  13th position: Q1 = 13.7% (Iowa) Q3:  75  i= ÷50 = 37.5  100  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% - 37 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter 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 fivenumber summary is 9.6, 13.7, 16.85, 20.2 and 29.95 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 New Hampshire Region 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 SE 12.2 Massachusetts NE 12.4 North Dakota MW 13.0 - NE 38 13.2 Virginia Vermont Poverty % © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures Generally, these states are the states with better economic conditions and less poverty The Northeast region with of the 12 states in this quartile appears to be the best economic region of the country The West region was second with of the 12 states in this group 66 a x= Σxi 4368 = = 364 rooms n 12 b y= Σyi 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 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 ( 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 ) 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 - 39 ( yi − y ) 1,764 13,689 16,384 1,444 1,444 31,684 31,684 19,044 8,100 47,524 1,369 174,13 ( 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 -74,35 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter sxy = Σ( xi − x )( yi − y ) −74,350 = = −6759.91 n −1 11 sx = Σ( xi − x ) 369, 074 = = 183.17 n −1 11 sy = Σ( yi − y ) 174,134 = = 125.82 n −1 11 rxy = s xy sx s y = −6759.91 = −.293 (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 67 a The scatter diagram is shown below - 40 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures 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 68 a - 41 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Chapter xi yi ( xi − x ) ( yi − y ) ( xi − x ) ( yi − y ) ( xi − x )( yi − y ) 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 -.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 sxy = sx = Σ( xi − x ) 1617 = = 1115 n −1 14 − sy = Σ( yi − y ) 0623 = = 0692 n −1 14 − rxy = b Σ( xi − x )( yi − y ) 0287 = = 0022 n −1 14 − 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 69 70 x= Σwi xi 20(20) + 30(12) + 10(7) + 15(5) + 10(6) 965 = = = 11.4 days Σwi 20 + 30 + 10 + 15 + 10 85 fi Mi f i Mi Mi − x - 42 ( Mi − x )2 fi ( Mi − x )2 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Descriptive Statistics: Numerical Measures 10 40 150 175 75 15 10 475 47 52 57 62 67 72 77 a x= 28,825 = 60.68 475 b s2 = 14,802.64 = 31.23 474 470 2080 8550 10850 5025 1080 770 28,825 -13.68 -8.68 -3.68 +1.32 +6.32 +11.32 +16.32 187.1424 75.3424 13 5424 1.7424 39.9424 128.1424 266.3424 1871.42 3013.70 2031.36 304.92 2995.68 1922.14 2663.42 14,802.64 s = 31.23 = 5.59 - 43 © 2010 Cengage Learning All Rights Reserved May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part ... -0.29 Total ( yi − y ) 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 .039 6 1.1960 0.0060 0.1840 0.6290 0 .032 2 0.2730 0.1189 2.4831 Σ( xi −... duplicated, or posted to a publicly accessible website, in whole or in part Chapter 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... -2.2607 0.1993 -0.7607 2.6076 0.9706 0.9706 1.0298 1.0298 6.1761 1.2428 1.0419 0. 9039 0.8663 1.7709 5.1109 0 .039 7 0.5787 1.6483 -0.9367 -0.9170 1.3505 2.2942 0.4952 0.8481 0.6593 0.2354 0.4346