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Chapter 03 - Descriptive Statistics: Numerical Methods

CHAPTER 3—Descriptive Statistics: Numerical Methods

3.1 a. Population parameter: number calculated using the population measurements that describe some aspect of the population.Point estimate: a one-number estimate of the value of a population parameter.

b. The sample mean is the point estimate of the population mean.

3.2 a. Equal or roughly equal.

b. Mean is less than the median.

c. Mean is greater than the median.

3.3 N MEAN MEDIAN MODE a 10 9.600 10.000 10.00b 9 103.33 100.00 90.00

3.4 N MEAN MEDIAN MODE a 10 20 20 20b 7 503 501 501

3.5 a. Sample mean = 42.954. Yes, because .

b. The sample median is 43.0The mean of 42.954 is slightly less than the median of 43.The histogram is slightly skewed to the left so the mean should be less than the median.

3.6 a. Sample mean = 5.46. Yes, because .

b. Sample median = 5.25The mean is slightly larger than the median. The histogram is somewhat skewed right which is consistent with the mean being larger than the median.

3.7 a. Sample mean = 50.575. Yes, because

b. Sample median = 50.65. The mean and median are very similar which is consistent with the histogram being close to symmetric.

3.8 Mean = 99 and the median = 40. Median is better measure of typical because of the high outlier (350).

3-1


3.9 – 3.13 MINITAB output for all variables on data comparing lifestyles.

a.

b.3.9 Somewhat skewed left; the U.S. has the lowest percentage.3.10 Slightly skewed left, the U.S. has the lowest percentage, tied with Britain.3.11 Skewed right; U.S. is the highest.3.12 Skewed right; U.S. is the highest.3.13 Skewed right; U.S. is above the mean and median.

c. All dot plots below show the same skewness as noted above.

3-2

Variable N Mean Median StDev Min Max Q1 Q3voters 9 70.70 71.50 11.81 49.1 85.0 61.35 80.40income tax 9 49.44 50.00 7.30 40.0 60.0 42.00 55.50video reentals 8 4.03 2.10 4.49 0.7 13.8 1.13 6.45PCs 8 18.94 17.50 7.10 11.5 35.0 14.50 20.00Religion 6 31.65 26.25 18.21 10.0 55.8 17.50 52.65


d. 3.9 U.S. has the lowest percentage.3.10 U.S. has the lowest percentage, tied with Britain.3.11 U.S. is the highest.3.12 U.S. is the highest.3.13 U.S. is above the mean and median.

3.14 Sample mean = 83.6 and sample median = 78.5 indicating data is skewed right.

3.15 a. Skewed to the right. Most players earned the league minimum.

b. Earning more than the mean: about 33%Earning more than the median: 50%

c. Explanations will vary. All quotes are a distortion/misrepresentation of the true earnings by the majority of players.

3.16 The range is the difference between the largest and smallest measurements in a population or sample. It is a simple measure of variation.

The variance is the average of the squared deviations from the average. It measures variability in square units.

The standard deviation is the square root of the variance.

3.17 Both the variance and standard deviation measure the spread of the individual values about the mean. The larger the spread, the more variation in the data.

3-3


3.18 The variance and standard deviation are calculated by using all of the data, while the range is calculated using only the largest and smallest values.

3.19 range = 15 – 5 = 10

3.20 Housing Affordability in Texas

x (x – 54) (x – 54)**2 sum(x – 54)**2 PopVariance PopStDev57.5 3.5 12.25 739.28 123.21 11.1061.7 7.7 59.2932.5 –21.5 462.2567.4 13.4 179.5655.7 1.7 2.8949.2 -4.8 23.04 Mean = 54

Range = 67.4 – 32.5 = 34.9Variance = 123.21Std Deviation = 11.10

3-4


3.21 a Airline Revenues

Revenue $ Billions (Rev - 11.43)^2

22.6 124.768919.3 61.936917.2 33.292913.1 2.788912.6 1.368911.6 0.02899.1 5.42893.3 66.09693.1 69.38892.4 81.5409

Sum 114.3 446.641

Mean = 11.43Variance = 446.641 / 10 = 44.664

Std. Dev. = Sqrt (Variance)= 6.683

Range = 22.6 – 2.4 = 20.200

Airline Profits

Profits$ millions (Rev - 1530.7)^2

231 1689220.0922,876 455621832.1-6,203 59810115.69

343 1410631.29-2,835 19059336.49

304 1504792.89499 1064404.89-53 2508105.69146 1917394.09

-1 2346104.89Sum 15307 546931938.1Mean = 1530.7Variance = 54693193.810Std. Dev. = 7395.485Range = 29079

3-5


b. United Airlines is a high outlier and the only observation above the average. All other airlines are within two standard deviations of the average.

Profits

Airline($

millions) z-scoreAmerican Airlines 231 -0.18United Airlines 22,876 2.89Delta Air Lines -6,203 -1.05Continental Airlines 343 -0.16Northwest Airlines -2,835 -0.59US Airways Group 304 -0.17Southwest Airlines 499 -0.14Alaska Air Group -53 -0.21SkyWest 146 -0.19Jetblue Airways -1 -0.21

3.22 a.

= 276.7

Also

b.

c. Yes, because $190 is not within the 99.73% interval.

d. z157= = 1.353

z132 = –0.150

z109 = –1.533

z145 = 0.631

3-6


z125 = –0.571

z139 = 0.271

Each expense is within 2 standard deviations of the mean.

3.23 a. The histogram indicates that the breaking strengths are approximately normally distributed. Therefore, the empirical rule is appropriate.

b. 68.26%: [50.575 ± 1(1.6438)] = [48.9312, 52.2188]95.44%: [50.575 ± 2(1.6438)] = [47.2875, 53.8626]99.73%: [50.575 ± 3(1.6438)] = [45.6436, 55.5064]

c. We estimate that the breaking strengths of 99.73% of the trashbags are between 45.6436 lbs and 55.5064 lbs. Thus, almost any trashbag will have a breaking strength that exceeds 45 lbs.

d. 27 out of 40 (or 67.5%) actually fall within the interval 38 out of 40 (or 95%) actually fall within the interval 40 out of 40 (or 100%) actually fall within the interval These percentages compare favorably with those given by the Empirical Rule (68.26%, 95.44%, and 99.73%). This suggests that our inferences are valid.

3.24 a. It is somewhat reasonable.

b.

c. Yes, because the upper limit of the 68.26% interval is less than 8 minutes.

d. 66% fall into 96% fall into , 100% fall into .Yes, they are reasonably valid.

3.25 a. It is somewhat reasonable.

b.

c. Yes, because the lower limit of the interval is greater than 35.

d. 63% fall into 98.46% fall into 100% fall into . Yes, they are reasonably valid.

3-7


3.26 a.

Stem and Leaf plot for Time stem unit = 10

leaf unit = 1

FrequencySte

m Leaf

27 3 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4

22 3 5 5 5 5 5 5 5 6 6 6 7 7 7 7 7 8 8 9 9 9 9 9

11 4 0 0 0 1 1 1 2 2 2 3 41 4 62 5 1 2

63

Histogram

0

5

10

15

20

25

30

35

32

34

36

38

40

42

44

46

48

50

52

54

Time

Per

cent

Distribution is skewed to the right.

3-8


b.

c. They are inconsistent with the empirical rule, but consistent with Chebyshev’s Theorem.

d. The transaction times are skewed to the right.

3.27 a. RS Internet Age Fund: [10.93 – 2*41.96, 10.93 + 2*41.96] = [–72.99, 94.85]Franklin Income A Fund: [13 – 2*9.36, 13 + 2*9.36] = [–5.72, 31.72]Jacob Internet Fund: [34.45 – 2*41.16, 34.5 + 2*41.16] = [–47.87, 116.77]

b. RS has the lowest average and highest variability. Franklin has the second lowest average and the smallest variability. Jacob has the highest average return and the second highest variability.

c. RS Internet Age Fund: Coefficient of Variation = 41.96 / 10.93 * 100 = 383.9%Franklin Income A Fund: Coefficient of Variation = 9.36 / 13 * 100 = 72%Jacob Internet Fund: Coefficient of Variation = 41.14 / 34.45 * 100 = 119.4%

RS Internet is riskiest, Jacob is second and Franklin is least risky.

3.28 a. The value such that a specified percentage of the measurements in a population or sample fall at or below it.

b. A value below which approximately 25% of the measurements lie

c. The 75th percentile

d. contains the middle 50% of the data.

3.29 In a box and whisker plot a point that is more than 1.5 times the IQR above Q3 or below Q1 is considered to be an outlier.

3.30 Median = 8. Q3 = 9. Q1 = 7.5.

3-9


3.31 a. 192 b. 152 c. 141 d. 171 e. 132 f. 30

g.

Income

(1000's) minimum 127 maximum 241 1st quartile 141.00 median 152.00 3rd quartile 171.00

BoxPlot

100 120 140 160 180 200 220 240 260 280

Income (1000's)

3.32 a. Explanations will vary.

b. High-tech sector is most variable. Retailing sector is least variable.

c. High-tech and Insurance sectors

3.33 Overall, the thirty-year rates are higher than the 15-year rates. The variability is similar.Average of the differences = .4444

3.34 a. Q1 = 7.5, Q3 = 9lower hinge = 7.5 (7 + 8) / 2 = 7.5upper hinge = 9 (9 + 9) / 2 = 9

b. Q1 = 138, Q3 = 177lower hinge = 139.5 (138 + 141) / 2 = 139.5upper hinge = 174 (171 + 177) / 2 = 174

c. explanations will vary

3.35 a. All categories showed improvement.

b. Most: strategic quality planning, quality and operational resultsLeast: information and analysis, human resource development and management

3-10


c. Increase: leadership, human resource development, customer focus and satisfactionDecrease: strategic quality planning, management of process qualityOthers stayed about the same.

d. The skewness changed in leadership, strategic quality planning, human resource development, and quality and operational results.

3.36 Both are measurements of linear association between two quantitative variables. If they are positive, then the relationship is positive and if they are negative, then the relationship is negative.

3.37 For a specific value of x you predict a value of y by plugging the value of x into the equation of the least squares line.

3.38 sxy = -179.6475 / 7 = -25.66, sx = sqrt(1404.355 / 7) = 14.16, sy = sqrt(25.548 / 7) = 1.91, r = -25.66 / (14.16 * 1.91) = -0.949, b1 = -25.66 / 14.162 = 0.1279, b0 = 10.2125 – (-0.1279) * 43.98 = 15.84

Predicted value at x = 40 degrees is 15.84 – 0.1279 * 40 = 10.724

3.39 a. There is a strong positive linear relationship between these two variables

b. Predicted service time is 11.4641 + 24.6022 * 5 = 134.4751 minutes.

3.40 Classess are weighted by credit hours. The xi values are the class grades and the weights are the credit hours per class.

3.41 The midpoint equals the average of the measurements in the class.

3.42 The actual measurements are unknown.

3.43 a.

b. unweighted mean = 10.72%

More money was invested in funds with larger gains.

3.44 a.

b. unweighted mean = 5.6%

3-11


The weighted mean accounts for the different sizes of labor force.

3.45 a. =

b. s2=

3.46

3.47 Midpoint Freq M*f (M – mean)**2 F*diff^2

30 1 30 462.25 462.2535 3 105 272.25 816.7540 3 120 132.25 396.7545 13 585 42.25 549.2550 14 700 2.25 31.555 12 660 12.25 14760 9 540 72.25 650.2565 1 65 182.25 182.2570 3 210 342.25 1026.7575 1 75 552.25 552.25

sample mean = 51.5 variance = 81.61017s = 9.033835

3-12


3.48 The geometric mean return for an investment is the constant return on the investment for the investment period.

3.49 Yes, the ending value is I(1+Rg)n , where I is the initial investment.

3.50 a.

b. $5,000,000(1+.0736)3=$6,187,500

3.51 1,000,000(1+Rg)4=4,000,000

(1+Rg)4 = 4

(1+Rg) =

Rg = 1.4142 – 1

Rg = .4142

3.52 a. 2000–01:

2001–02:

b. 2002–03:

2003–04:

2004–05:

c. Rg=

d. $1,000,000(1 – .036)5 = $832,501

3.53 a. 1990–95:

1995–2000:

Rg =

3-13


b.

3.54 Times greater than 17 are high outliers and they should be investigated.

BoxPlot

0 5 10 15 20 25 30

Time

3.55 Explanations will vary.

3.56 a. Fixed Annuities: 68.26% [ ± ] = [8.31 ± .54] = [7.77, 8.85]99.73% [ ± 3] = [8.31 ± 3(.54)] = [6.69, 9.93]

Cash Equivalents: 68.26% [ ± ] = [7.73 ± .81] = [6.92, 8.54]99.73% [ ± 3] = [7.73 ± 3(.81)] = [5.30, 10.16]

U.S. Treasury Bonds: 68.26% [ ± ] = [8.80 ± 5.98] = [2.82, 14.78]99.73% [ ± 3] = [8.80 ± 3(5.98)] = [–9.14, 26.74]

U.S. Corporate Bonds: 68.26% [[ ± ] = [9.33 ± 7.92] = [1.41, 17.25]99.73% [ ± 3] = [9.33 ± 3(7.92)] = [–14.43, 33.09]

Non U.S. Gov’t Bonds: 68.26% [ ± ] = [10.95 ± 10.47] = [.48, 21.42]99.73% [ ± 3] = [10.95 ± 3(10.47)] = [–20.46, 42.36]

Domestic Large Cap Stocks: 68.26% [ ± ] = [11.71 ± 15.30] = [–3.59, 27.01]99.73% [ ± 3] = [11.71 ± 3(15.30)] = [–34.19, 57.61]

International Equities: 68.26% [ ± ] = [14.02 ± 17.16] = [–3.14, 31.18]99.73% [ ± 3] = [14.02 ± 3(17.16)] = [–37.46, 65.5]

3-14


Domestic MidCap Stocks 68.26% [ ± ] = [13.64 ± 18.19] = [–4.55, 31.83]99.73% [ ± 3] = [13.64 ± 3(18.19)] = [–40.93, 68.21]

Domestic Small Cap Stocks 68.26% [ ± ] = [14.93 ± 21.82] = [–6.89, 36.75]99.73% [ ± 3] = [14.93 ± 3(21.82)] = [–50.53, 80.39]

b. and c. are for fixed annuities; others follow similar methods. The results for b are obtained by using Chebyshev’s Theorem for k=2 and k=3. The results for c are 3 standard deviations above the mean and 3 standard deviations below the mean.

b. At least 75%: [7.23, 9.39]; at least 88.89%: [6.69, 9.93]

c. 9.93, 6.69

d. Domestic small cap stocks: 80.39, domestic midcap stocks: 68.21

e. Fixed annuities: 6.69; cash equivalents: 5.30

f. Least risky: fixed annuities coefficient of variation = (.54/8.31)100 = 6.498

Riskiest: domestic small cap stockscoefficient of variation = (21.82/14.93)100 = 146.149

3.57 a. About 70%

b. 410

3.58 a. 538 3(41) = 538 123 = [415, 661]

b. Yes, it is beyond the interval.

3.59 a. Average Price with Pools = $276,056

Average Price without Pools = $226,900Difference = $49,156% Recouped = $49,156/32,500 = 151%

b. The assumption that the only variable causing the difference in average price is whether or not there is a pool is really suspect.

Internet Exercise

3.60 Analyses will vary

3-15

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Duffy Template - Faculty Web Sites - Texas A&M …faculty.tamuc.edu/emanton/gbus578/Solutions/Chap003.doc · Web viewLast modified by jaikumar.v Created Date 7/30/2007 6:38:00 AM