ch1 - 1.6 1.8 1.18 1.20 1.26 2.4 Quantitative data are...

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Unformatted text preview: 1.6 1.8 1.18 1.20 1.26 2.4 Quantitative data are measurements that are recorded on a meaningful numerical scale. Qualitative data are measurements that are not numerical in nature; they can only be classified into one of a group of categories. A population is a set of existing units such as people. objects, transactions, or events. A sample is a subset of the units of a population. a. Since the members were asked to fill out a questionnaire, the method of data collection is a mail survey. The population surveyed is the 4000 members of the Institute of Management. The variable of interest is the expected change in the number of temporary employees by 2002. Since there were only 3 possible choices for answers to the question that are mnnumerical, the variable is qualitative. Since fewer than half of the members in the sample indicated that they expected to increase the number of temporary employees by 2002, we can infer that fewer than half of all members expect to increase the number of temporary employees by 2002. The variable of interest to the researchers is the rating of highway bridges. Since the rating of a bridge can be categorized as one of three possible valueI. it is qualitative. The data set analyzed is a population since all highway bridges in the U.S-. were categorized. The data were collected observationally. Each bridge was observed In in mural setting. The pepulation of interest is the set of all adults in the Minhang District, a suburb of Shanghai, China. The sample size was 3,423 + 3,593 = 7,016. The study made inferences about all "people in China." Since pnly those in the Minhang District were sampled, the results may not be characteristic of all Chinese people. The group of people surveyed was from a very small group of people in China.‘ The people in the Minhang District may be quite different from the population of China in general. To construct a relative frequency table for the data, we must find the relative frequency for each Cruise line. To find the relative frequency, divide the frequency by the total population size, 1,591,560. The relative frequency for Canaveral is 152,240r’1,591,560 = .096. The rest of the relative frequencies are found in a similar manner and are reported in the table. 2.8 2.12 ' (frutse Elite Number oi Passengers fielanve Frequency (fanaveral (Dolphmi 152.240 15224011591350 = .133 Carnival (Fantasy) 480,924 4809241 1591560 = .302 Disney (Magic) 73,504 7350411591560 =1 .046 Premier (Oceanic) 270,361 27036111591560 = .170 Royal Caribbean (Nordic Empress) 106.161 10616111591560 = .067 Sun Cruz Casinos 453,806 45380611591560 = .285 Sterling Cruises (New Yorker) 15.782 1578211591560 = .010 Topaz Int’l Shipping (Topaz) 28,820 2828011591560 = .018 Other 10.502 1050211591560 = .007 oral 1,591, 1. 1 b. The cruise ship with the highest relative frequency (.302) is Carnivai (Fantasy). This means - that Carnival handles approximately .302 of the total pepulation. The relative frequency bar chart is: 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 fife fa dfqéeifczyd’” fag as Most of the books (63%) received a "favorableirecomrnended" review. About the same percentage of books received the following reviews: "cautious or very little recommendation" (10%), "little or no preference" (9%), and "outstandingi’significant contribution" (12%). Only 5% of the books received "would not recommend" reviews. If the top two categories are added together. the percent recommended is "15% (actually slightly higher than "15%). This agrees with the study. __.. 1‘ fl Rainw- firm ' is . ..__l_ a . ..__1__ .5 as 4.5'55 as 113.5 12514.5 use “mamas: L. 16 b. The highest proportion of test scores {.25} fell in the measurement class 15—95 The proportion of scores between 3.5 and 5.5 is .15 The proportion of test scores higher than 11.5 is .10 + .05 + .05 = .20 The proportion of the 100 Students who scored less than 5.5 is .05 + .15 = .20 Using MINKTAB, the stem-and-leaf display is: Steward-leaf of PAF N=17 Leaf Unit = 1.0 6 0 000009 8 ‘l 25 (2) 2 1+5 ? 3 13 5 4 0 4 5 4 6 2 3 F 05? The median is the middle number once the data are arranged in order. The data arranged in order are: 0.0,0,0,0,9,12, l5. 24, 25, 31, 33. 40, 6'2, 70, 75. "IT. The middle number or the median is 24. 77+33+T5+---+31 4'53 17 l? = 27.82 The mean of the data is f 2 £3: = H The number occurring most frequently is 0. The mode is 0. The mode corresponds to the smallest number. It does not seem to locate the center of the distribution. Borh the mean and the median are in the middle of the stem—and-leaf display. Thus, it appears that both of them locate the center of the data. “L: H a ll H s = m = 1.1402 After adding 3 to each of the data points, Range = 6 — 3 = 3 222 x2— (BX): 102—... s2=_£—”—_5=1.3 = nwl 5—1 l m =1.1402 After subtracting 4 from each of the data points, Range = —1—(—4)= 3 2 _ 2 2x2_ (Ex) 39 _(13) 52: '1 =____._L :13 3= 1.3 =1.1402 11—1 5—1 The range, variance. and standard deviation remain the same when any number is added to or subtracted from each measurement in the data set. 2.60 From the printout, the mean is 40.0555556 and the standard deviation is 2.1770812. Both of these measures are measured in the same units as the original data, which is miles per gallon. Since the sample mean is a good estimate of the population mean, the manufacturer should be satisfied. The sample mean is 40.0555556 which is greater than 40. The range of the data set is 45 — 35 = 10. Using Chebyshev’s Rule, the range should cover approximately 6 standard deviations. Thus, a good estimate of the standard deviation would be 1016 = 1.67. Using the Empirical Rule, the range should cover approximately 4 standard deviations. Thus, a good estimate of the standard deviation would be 1074 = 2.5 The given standard deviation is 2.2 which is between these two estimates. Thus, it is a reasonable value. Using MINITAB, the frequency histogram is {the relative frequency histogram would have the same shape): Histogram of [:1 It 36 Midpoint 35 36 37 38 39 40 41 42 (.3 u. 45 i I 'I' *i* tit ii ti- lt Ifi'fifi *‘kt t**t*I* “111' it i I d—hNb‘fi‘ObUU—‘J Yes, the data appear to be mound-shaped. Because the data are mound-shaped, we can use the Empirical Rule. We would expect approximately 68% of the data within the interval Y i 3, approximately 95% of the data within the interval 3 i 2.5, and approximately all of the data within the interval Y i 35. The interval E i s is 40.056 i 2.177 or (37.879, 42.233). Twenty-seven of the observations fall in this interval or 27136 = .75 or 75%. This number is a little larger than 68%. The interval E i 25 is 40.056 1 20.177) or (35.702, 44.410). Thirty-four of the observations fall in this interval or 34736 x .94 or 94%. This number is very close to 95%. The interval .Y i 35 is 40.056 i 3(2. 177) or (33.525, 46.587). Thirty-six of the observations fail in this interval or 36736 = 1.00 Or 100%. This number is the same as all of the observations. 2.72 Since the element 40 has a z-score of —2 and 90 has a z-score of 3, —2= 40—pand3=90_’u U U =--20=40—p. =30=90—p un—zamo =p+30=90 I» p, = 40 + 20 By substitution, 40 + 20 + 30 = 90 w 50 = 50 =9 o = 10 By substitution, p, = 40 + 2(10) = 60 Therefore, the population mean is 60 and the standard deviation is 10. 2.78 a. From the problem, .u. = 2.7 and a = .5 z= x ;“=zo=x—p=>x:p+zcr Forz = 2.0, x = 2.? + 2.0(.5) = 3.7 For 2: = —1.0,x = 2.7 - l.0(.5) = 2.2 For 2: = .5, x = 2.7 + .5(.5) : 2.95 Forz = —2.5,x = 2.7 — 2.5(.5) = 1.45 b. For 2 = —3.6,x = 2.7 —1.6(.5)= 1.9 e. If we assume the distribution of GPAs is approximately mound~shaped. we can use the Empirical Rule. From the Empirical Rule, we know that z .025 or = 2.5 % of the students will have GPAs above 3.7 (with z = 2). Thus, the GPA corresponding to summa cum laude (top 2.5%) will be greater than 3.? (z > 2). We know that :16 or 16% of the students will have GPAs above 3.2 (z = 1). Thus, the limit on GPAs for cum laude (top 16%) will be greater than 3.2 (2 > 1). We must assume the distribution is mound-shaped. ...
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This note was uploaded on 02/28/2008 for the course ECON 317 taught by Professor Safarzadeh during the Fall '07 term at USC.

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ch1 - 1.6 1.8 1.18 1.20 1.26 2.4 Quantitative data are...

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