46 Chapter 8 Statistics: An Introduction

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1 46 Chapter 8 Statistics: An Introduction Activity 5 Continued Box 4 1. The median is 6. The mode is 5. The mean is about 7 (6.8). 2. The median and the mode were unchanged, but the mean increased significantly. The length of 13 letters is very large in comparison to the other lengths. The mean of a set of data may, as this example shows, be greatly affected by very large or very small data points. 3. The median is 6.5. The mode is 8. The mean is about 6.5 (6 7/12). 4. All three averages were increased by these additions. The mean and the median increased about the same amount. The mode was affected the most, even though 8 occurred only one more time than 5. The mean always tends toward the mode whereas the median may either be moved toward the mode or away from it depending on whether the mode is above (as in this case) or below the center of the data. 5. To compute the mean, add the lengths of the names and divide the total by n. WHICH WOULD YOU USE? 1. Sam would use the mean because the.495 batting average makes it largest. 2. The owner would use the mode because it is the smallest average. 3. An arbitrator would probably use the median because.495 is unusually large in comparison to the rest of the data (thus the mean would not accurately reflect the data) and 2 out of 5 is not an unusually high frequency for the mode. IDENTIFY THE AVERAGE The average shoe size is probably the mode. It is likely that one shoe size would be sold with a higher frequency than the others. This would affect both the median and the mean. The average household is the mean. Household sizes tend to fall in a narrow range and no single size occurs with a significantly greater frequency than the others. Also, since household sizes are whole numbers, the mode would also be a whole number, and the median would either be a whole number or the decimal portion of the number would be.5. The average household income is probably the median. The mean would be greatly affected by very high or very low incomes, and it is unlikely that there is a single income that occurs with an unusually high frequency that would affect the median. Activity 6 Suggestions: The data for this activity should be collected during class. The remainder of the activity may be completed in or out of class. The answers to most exercises depend on the data collected in the activity. 14. a. About 25% of the data points lie between the lower extreme and the lower quartile. b. About 25% of the data points lie between the upper extreme and the upper quartile. c. About 25% of the data points lie between the lower quartile and the median. d. About 25% of the data points lie between the median and the upper quartile. 15. About 50% of the data points lie in the box.

2 Chapter 8 Answers 47 Activity 6 Continued EXTENSIONS 2. b. The extremes, the range, and the median can all be readily identified in both stem-andleaf and box-and-whisker plots. c. Outliers and quartiles are more easily identified in a box-and-whisker plot than in a stem-andleaf plot. Data sets, especially ones of different sizes, are more easily compared using box-andwhisker plots. d. The mode and gaps and clusters in the data can be identified in a stem-and-leaf plot, but not in a box-and-whisker plot. Activity 7 Suggestions: It would be helpful if students complete Activity 3: Grouped Data, which develops stemand-leaf plots, and Activity 4: What s the Average?, which develops the concepts of mean and median and explores the effects of extremes in data, before doing this activity. Activity 6 will extend and reinforce the concepts explored in Activities 3 and 4. Discussion of Activity 6 should focus on analyzing the various plots, determining what information can be more easily derived from one display than another, and evaluating the use of mean or median as a good descriptor of average. 1. San Francisco Wichita 2. San Francisco Wichita Mean Median Mode 61, 64 none J F M A M J J A S O N D 6. 2 months 7. the line graph; The box plots do not show the individual temperatures. 8. The mean and median temperatures are about the same, but the range of the temperatures varies widely. 9. Neither the mean nor median temperature accurately describes the annual temperature for Wichita since the range is so large; either the mean or median accurately describes the climate in San Francisco since the range is only 15 degrees.

3 48 Chapter 8 Statistics: An Introduction Activity 7 Continued 10. The range of data 11. Wichita 37 43' NSan Francisco 37 47' N San Francisco is further north 12. San Francisco is on the coast so the climate is moderated by the Pacific Ocean currents. 13. Portland, ME Portland, OR Portland, OR: Latitude 45 31' N, Mean = 53.5, Median = 52.5, Mode = 40 Portland, ME: Latitude 43 40' N, Mean = 45.9, Median = 46, there is no mode. For seven months of the year, the temperature in Portland, ME, is within the range of temperatures in Portland, OR. Portland, OR, is further north, but its climate is moderated by the Pacific Ocean currents J F M A M J J A S O N D 15. Answers will vary.

4 Chapter 8 Answers 49 Activity Women Women Men Men Both the men's and women's times for the 200-meter run are decreasing slightly, but the difference between them is staying constant. 4. Both the men's and women's times for the 200-meter run are decreasing, and the women's time appears to be decreasing at a slightly faster rate. 5. The conclusions are probably not the same. In Exercise 2, it appears that women's times are not catching up to the men's times. In Exercise 4, it appears that women are catching up. 6. The change in scale on the vertical axis magnifies the differences between the men's and women's times so that small changes are more apparent See graph on next page. 12. a. about 2072 b. It is about 15 years later than predicted by the researchers. c. about 17.8 sec 13. Possibly because more women have begun running competitively in recent decades, and because their training has become more intense in the past 10 to 15 years. 14. Answers will vary, but should include an interpretation of the data in this activity. 15. Answers will vary. Again, responses should include an interpretation of the data in this activity and may include arguments related to women's generally smaller stature and anatomical differences such as hip width. EXTENSIONS Answers will vary.

5 50 Chapter 8 Statistics: An Introduction Activity 8 Continued Activity Answers will vary. One approach is to divide the scores into two groups (those that have a school year less than the median of 191 days, and those that have a school year greater than or equal to the median) and construct box-and-whisker plots for the two groups. This analysis indicates that countries with longer school years tend to have higher math scores. A similar analysis using the median of the instruction time (289 min) indicates that math performance is independent of the number of minutes of instruction.