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be calculated for quantitative data only, and which can be calculated for both quan- titative and qualitative data? Illustrate with examples. 3.5 Which of the five measures of center (the mean, the median, the trimmed mean, the weighted mean, and the mode) can assume more than one value for a data set? Give an example of a data set for which this summary measure assumes more than one value. 3.6 Is it possible for a (quantitative) data set to have no mean, no median, or no mode? Give an example of a data set for which this summary measure does not exist. 3.7 Explain the relationships among the mean, median, and mode for symmetric and skewed histograms. Illustrate these relationships with graphs. 3.8 Prices of cars have a distribution that is skewed to the right with outliers in the right tail. Which of the measures of center is the best to summarize this data set? Explain. 3.9 The following data set belongs to a population: 5 −7 2 0 −9 16 10 7 Calculate the mean, median, and mode. APPLICATIONS 3.10 The following data give the 2014 profits (in millions of dollars) of the top 10 companies listed in the 2014 Fortune 500 (source: www.fortune.com). Company 2014 Profits (millions of dollars) Wal-Mart Stores 16,022 Exxon Mobil 32,580 Chevron 21,423 Berkshire Hathaway 19,476 Apple 37,037 Phillips 66 3726 General Motors 5346 Ford Motor 7155 General Electric 13,057 Valero Energy 2720 Find the mean and median for these data. Do these data have a mode? Explain. 3.2 Measures of Dispersion for Ungrouped Data 89 139 151 138 153 134 136 141 126 109 144 111 150 107 132 144 116 159 121 127 113 a. Calculate the mean, median, and mode for these data. b. Calculate the 10% trimmed mean for these data. 3.19 In a survey of 640 parents of young children, 360 said that they will not want their children to play football because it is a very dangerous sport, 210 said that they will let their children play football, and 70 had no opinion. Considering the opinions of these parents, what is the mode? 3.20 A statistics professor based her final grades on quizzes, in-class group work, homework, a midterm exam, and a final exam. However, not all of the assignments contributed equally to the final grade. John received the scores (out of 100 for each assignment) listed in the table below. The instructor weighted each item as shown in the table. Assignment John’s Score (Total Points) Percentage of Final Grade Assigned by the Instructor Quizzes 75 30 In-class group work 52 5 Homework 85 10 Midterm exam 74 15 Final exam 81 40 Calculate John’s final grade score (out of 100) in this course. (Hint: Note that it is often easier to convert the weights to decimal form for weighting, e.g., 30% = .30; in this way, the sum of weights will equal 1, making the formula easier to apply.) 3.21 A clothing store bought 8000 shirts last year from five different companies. The following table lists the number of shirts bought from each company and the price paid for each shirt. Company Number of Shirts Bought Price per Shirt (dollars) Best Shirts 1200 30 Top Wear 1900 45 Smith Shirts 1400 40 New Design 2200 35 Radical Wear 1300 50 Calculate the weighted mean that represents the average price paid for these 8000 shirts. 3.22 One property of the mean is that if we know the means and sample sizes of two (or more) data sets, we can calculate the com- bined mean of both (or all) data sets. The combined mean for two data sets is calculated by using the formula Combined mean = x = n1x1 + n2x2 n1 + n2 where n1 and n2 are the sample sizes of the two data sets and x1 and x2 are the means of the two data sets, respectively. Suppose a sample of 10 statistics books gave a mean price of $140 and a sample of 8 math- ematics books gave a mean price of $160. Find the combined mean. (Hint: For this example: n1 = 10, n2 = 8, x1 = $140, x2 = $160.) 3.23 For any data, the sum of all values is equal to the product of the sample size and mean; that is, ∑x = nx. Suppose the average amount of money spent on shopping by 10 persons during a given week is $105.50. Find the total amount of money spent on shopping by these 10 persons. 3.24 The mean 2015 income for five families was $99,520. What was the total 2015 income of these five families? 3.25 The mean age of six persons is 46 years. The ages of five of these six persons are 57, 39, 44, 51, and 37 years, respectively. Find the age of the sixth person. 3.26 Seven airline passengers in economy class on the same flight paid an average of $361 per ticket. Because the tickets were purchased at different times and from different sources, the prices varied. The first five passengers paid $420, $210, $333, $695, and $485. The sixth and seventh tickets were purchased by a couple who paid identical fares. What price did each of them pay? 3.27 When studying phenomena such as inflation or population changes that involve periodic increases or decreases, the geometric mean is used to find the average change over the entire period under study. To calculate the geometric mean of a sequence of n values x1, x2,..., xn, we multiply them together and then find the nth root of this product. Thus Geometric mean = n √x1 · x2 · x3 · . . . · xn Suppose that the inflation rates for the last five years are 4%, 3%, 5%, 6%, and 8%, respectively. Thus at the end of the first year, the price index will be 1.04 times the price index at the beginning of the year, and so on. Find the mean rate of inflation over the 5-year period by finding the geometric mean of the data set 1.04, 1.03, 1.05, 1.06, and 1.08. (Hint: Here, n = 5, x1 = 1.04, x2 = 1.03, and so on. Use the x1/n key on your calculator to find the fifth root. Note that the mean inflation rate will be obtained by subtracting 1 from the geometric mean.) * * * * * * The measures of center, such as the mean, median, and mode, do not reveal the whole picture of the distribution of a data set. Two data sets with the same mean may have completely different spreads. The variation among the values of observations for one data set may be much larger or smaller than for the other data set. (Note that the words dispersion, spread, and variation have similar meanings.) Consider the following two data sets on the ages (in years) of all workers at each of two small companies. Company 1: 47 38 35 40 36 45 39 Company 2: 70 33 18 52 27 3.2 Measures of Dispersion for Ungrouped Data 90 Chapter 3 Numerical Descriptive Measures The mean age of workers in both these companies is the same, 40 years. If we do not know the ages of individual workers at these two companies and are told only that the mean age of the work- ers at both companies is the same, we may deduce that the workers at these two companies have a similar age distribution. As we can observe, however, the variation in the workers’ ages for each of these two companies is very different. As illustrated in the diagram, the ages of the workers at the second company have a much larger variation than the ages of the workers at the first company. Company 1 36 39 35 38 40 45 47 Company 2 18 27 33 52 70 Thus, a summary measure such as the mean, median, or mode by itself is usually not a suf- ficient measure to reveal the shape of the distribution of a data set. We also need a measure that can provide some information about the variation among data values. The measures that help us learn about the spread of a data set are called the measures of dispersion. The measures of center and dispersion taken together give a better picture of a data set than the measures of center alone. This section discusses four measures of dispersion: range, variance, standard deviation, and coef- ficient