A C E. Answers Investigation 3. Applications. Sample 2: 11 moves. or 0.44; MAD Sample 2: 22. , or 2.44; MAD Sample 3: 0, or 0.

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1 Applications 1. a) The range is $1.75. b) Each server receives $ c) Since Yanna s amount is higher than the mean, they will each receive more. If Yanna receives the mean ($15.65), then the remainder of her tips ($.45) is shared among the five people. Each receives an additional 9 cents. 2. The mean is an average of all the data values, so it is entirely possible that no one received the actual mean. 3. a. Yanna could have made a low amount of tips, but the other servers made high tips, so the result would be a mean of $ b. The information is incomplete, but we know that Yanna has a smaller amount of tips than the mean, so this suggests that there must have been larger tips to offset this amount in order to make the mean $ a. IQR Korean names: 3 letters (from 8 to 11 letters) IQR U.S. names: 3 letters (from 11 to 14 letters) IQR Japanese names: 2 letters (from 10 to 12 letters) IQR Chinese names: 2 letters (from 6 to 8 letters) b. Both Japanese and Chinese name lengths are the least spread out with the smallest IQRs. The U.S. and Korean name lengths are the most spread out, so they have greater variability. 5. a. Sample 1: 2 moves Sample 2: 11 moves b. The mean household size is the number that each household would have if the people were distributed so that each household has the same number. The evened-out bar graph shows the households as having an equal number of people, so it shows the mean. c. In Sample 3, the data vary the least from the mean; in Sample 2 the data vary the most from the mean. It takes 0 moves to even out the bars of Sample 3; the bars are already evened out. It takes the most moves to even out the bars of Sample 2. d. MAD Sample 1: 4 9 or 0.44; MAD Sample 2: 22 9, 4 2 9, or 2.44; MAD Sample 3: 0, or 0. 9 Sample 2 varies the most from the mean of 5 people; Sample 3 varies the least from the mean of 5 people. This is because greater MADs indicate that data vary more from the mean. Lesser MADs indicate that data don t vary that much from the mean. Data About Us 1

2 6. a. (See Figures 1 and 2.) b. Jeff: median: 80, IQR; 7.5; Elaine: median: 80, IQR: 20 c. Jeff: mean: 80, MAD: 3.75; Elaine: mean: 80, MAD: 10 d. Jeff; Both have the same median and mean scores; however, Elaine s data show greater variability, so that means that she is less likely than Jeff to receive an exact score of a. Ride 1: mean is 15 minutes Ride 2: mean is 18 minutes b. Ride 1: MAD is 2.4 minutes Ride 2: MAD is 8 minutes c. The data vary more for Ride 2. Ride 2 has a greater MAD, which indicates greater variability. d. Answers will vary based on the data sets that students use. 8. Distribution A: MAD is about 1.49, and IQR is 3. Distribution B: MAD is about 2.2, and IQR is 3. Distribution C: MAD is 0.96, and IQR is C has the least variation from the mean. B has the most. 10. A and B have the same spreads. C has the least spread. 11. a. Figure 1 Figure 2 Data About Us 2

3 b. Class 1: median: 2 pets, IQR: 4 pets; Class 2: median: 2.5 pets, IQR: 2.5 pets; Class 3: median: 3.5 pets, IQR: 6 Possible answers: The medians are close for all classes, but Class 3 has the largest median. Class 3 has the highest IQR, so its data are more spread out than the data for the other classes. Using IQR, Class 1 data is slightly more spread out than the data for Class 2. c. Possible answers: Class 3 has the largest mean. The distribution of number of pets for Class 3 is more spread out than for the other two classes. Connections 12. a. Malaika would need a score of 19 on her fourth project. Students might say that the mean is the height all the bars would be if Malaika received the mean score on all her projects. Given the mean of 17 points, all four bars would be at a height of 17, for a total of 68 points. Right now, Malaika has a total of 49 points, so she needs the fourth project to score 19 points. b. The spread is 15 to 19 points, for a range of 4 points. Some students might argue that there is not much variation in her scores. Others may convert scores to percentages (i.e., 15 points is 75%, and 19 points is 95%) that they interpret as grades and decide that there is, indeed, more variation, as the impact on final grades definitely matters. 13. a. Her mean score would be 20 points; 15 points. b. If Malaika has a total of 80 points, then she must have scores of 20 on each project. For a total of 60 points, there are many options, such as 20, 10, 10, and 20. c. For a total 80 points, each project will have a score of 20, so the range is 0. For a total of 60 points, the range depends on the student s data set in part (b). For the example above, the range would be 10 (20 10). d. For a total of 80 points, the range is less variable (no variation) that her first set of scores. For a total of 60 points, the variability depends on the student s data set in part (b) compared to the answer for Exercise 12, part (b). 14. a. mean = about 26.4, median = 26 b. mean = about 37.8, median = 25 c. Other drinks most likely have more variability since the mean and median are further apart. The ranges also indicate that there is more variability in other drinks than in soda drinks. d. Possible answers: There is a larger range of caffeine in other drinks than in soda drinks. The medians of soda drinks and other drinks are almost equal. Since the mean is larger than the median in the other drinks, the line plot of the other drinks data would probably be skewed to the right, whereas the line plot of the soda drinks data would most likely be symmetrical. 15. a. true b. false c. false; 50, have 25 mg or less 16. a. MAD = 5.16 mg and IQR = 10 mg best describe the variability in the caffeine content of soda drinks. The overall range for soda drinks is 23, so the IQR cannot be greater than that. Data About Us 3

4 b. MAD = mg and IQR = 59 mg best describe the variability in the caffeine content of the other drinks. The range of the caffeine content in the other drinks (81 mg) is much greater than the range of the caffeine content in soda drinks. So, the other measures of variability will be greater for other drinks as well. 17. Both the mean and the median name length for female names is greater than the measures of center for male names. So, female names are longer. 18. a. Student answers may vary. Sample dot plot: b. The change in data values will shift the median to a smaller value. c. The change will affect the mean; the higher values being removed and the four values that are less than the mean being added in will cause the mean to be less than it was originally. 19. B 20. J 21. A 22. a. The mean is approximately J 24. D 25. G b. The mean is multiplied by the same factor as the individual scores. So, The means are about 16.53, about 5.51, and about 1.65 for the given situations. c. The response of the mean to multiplying each score by the same factor is an application of the distributive property. For example, if the 6 scores are a, b, c, d, e, f, then the mean M = (a + b + c + d + e + f) 6. If you multiply each score by a factor X, then the scores are Xa, Xb, Xc, Xd, Xe, Xf, and the new mean is (Xa + Xb + Xc + Xd + Xe + Xf) 6 = X(a + b + c + d + e + f) 6 = XM. The mean grows by a factor of X. Extensions 26. Yes; since the definition of mean implies that you should divide the sum of the scores by the total amount of scores. This shows the Distributive Property of Multiplication Over Addition. Rather than multiplying the sum by one half or one third, Mark is multiplying each addend by one half or one third. 27. a. mode = 14, mean = 12.2, median = 13, IQR = 7, MAD = 3.6 b. mode = 17, mean = 15.2, median = 16, IQR = 7, MAD = 3.6 Since all values are shifted the same amount, the measures of center all shift by the same amount. For example, if you add 3 to each data value, you will add 3 to the mean, mode, and median. However, the measures of variability remain the same. The IQR measures how different Q1 and Q3 are. Since both are 3 more than before, the difference stays the same. The MAD measures each data value s distance from the mean and then finds the average. Since each value is 3 higher than before, and the mean is also 3 higher, the distances from the mean are unchanged, so the MAD is unchanged. The distribution has just been moved right on the axis; it has the same shape as before. Data About Us 4

5 c. m c. mode = 28, mean = 24.4, median = 26, IQR = 14, MAD = 7.2 Since all values are affected by the same multiplicative values, the measures of center and variability will feel the same affect. If you multiply each data value by 2, you will multiply the mean, mode, median, IQR, and MAD by 2. Data About Us 5