# Homework 8 Solutions

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2 7. Comparing Earnings. Figure 5. compares the average weekly earnings of men and women. Identify any misleading aspects of the display. Draw the display in a fairer way. Average weekly earnings \$9 \$8 \$7 \$6 \$5 \$ \$3 \$ \$ \$ Men Women The graph does not start at zero, making it appear that women earn about a quarter of what men earn. A fairer graph would start from zero. 8. Breaking Distances. Figure 5. shows the breaking distance for three different cars. Discuss the ways in which it might be deceptive. How much greater is the breaking distance of a Lincoln than the breaking distance of a Lexus? Draw the display in a fairer way. As presented in Figure 5., the Lincoln superficially appears to have a breaking distance about twice that of the Lexus because the x-scale starts at 7 feet. In fact, the difference is only about (7 87) = feet, or (7 87)/87 =.7%. A more fair way of representing the data might start the graph at zero: Lincoln Saab Lexus Breaking distance (feet) 9. Cell Phone Users. The following table shows the number of cell phone subscribers in the United States for selected years between 99 and 7. Display the data using both an ordinary vertical scale and an exponential vertical scale. (Hint: For the exponential scale, use tick marks at million, million, and million.) Which graph is more useful? Why? Year Subscribers (millions)

3 Ordinary Scale Exponential Scale Subscribers (millions) Subscribers (millions) Either scale has its uses, the ordinary scale shows a steadily increasing growth, while the exponential scale illustrates that this growth is somewhat less than exponential. Chapter 6A Does it make sense? Decide whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning.. The distribution of grades was left-skewed, but the mean, median, and mode were all the same. Does not make sense. If the mean, median, and mode are the same, the distribution should be symmetric. Mean, Median, and Mode. Compute the mean, median, and mode of the following data sets.. Body temperatures (in degrees Fahrenheit) of randomly selected normal and healthy adults: The mean, median, and mode are 98., 98., and 98., respectively.. Margin of Victory. The data set below gives the margin of victory in the NFL Superbowl games for a. Find the mean and median margin of victory. The mean is.5 points and the median is 7 points. b. Identify the outliers in the set. If you eliminate the outliers on the high side, what are the new mean and median. After eliminating outliers on the high side, the mean is 5.83 points and the median is 3 points. Approximate Average. State, with an explanation, whether the mean, median, or mode gives the best description of the following averages. 3. The average number of times that people change jobs during their careers. The distribution is probably right-skewed by a few people who change jobs frequently, so the median will give a better description.

4 Describing Distributions. Consider the following distributions. a. How many peaks would you expect from the distribution? Explain. b. Would you expect the distribution to be symmetric, left-skewed, or right-skewed? Explain. c. Would you expect the variation of the distribution to be small, moderate, or large? Explain. 7. The exam scores for students when students got an F, 5 students got a D, and students got a C. a. There would be one peak on the far left (F s). b. The distribution would be right-skewed because the scores trail off to the right. c. The variation is large. 3. The weights of cars at a dealership at which about half of the inventory consists of compact cars and half of the inventory consists of sport utility vehicles. a. There would likely be two peaks, one for compact cars, and one for sport utility vehicles. b. The distribution would be symmetric. c. The variation would be moderate because, although the difference in weight between compact cars and sport utility vehicles is large, the differences between compact cars or between sport utility vehicles tends to be small. Smooth Distributions. Through each histogram, draw a smooth curve that captures its important features. Then classify the distribution according to its number of peaks, symmetry or skewness, and variation. 35. Times between 3 eruptions of Old Faithful geyser in Yellowstone National Park, shown in Figure 6.6. Times Between Eruptions of Old Faithful 6 5 Frequency Time (minutes) The distribution has two peaks (i.e., it is bimodal), with no symmetry and large variation.

5 36. Time until failure for a sample of 8 computer chips that failed, shown in Figure 6.7. Frequency Failure Time of Computer Chips Time (months) The distribution has one peak and is right-skewed. Although most of the data is clustered around its peak, the distribution has considerable spread, so it has moderate variation. 39. Family Income. Suppose you study family income in a random sample of 3 families. You find that the mean family income is \$55,; the median is \$5,; and the highest and lowest incomes are \$5, and \$, respectively. a. Draw a rough sketch of the income distribution, with clearly labeled axes. Describe the distribution as symmetric, left-skewed, or right-skewed. Sketches will vary, but, because the mean is larger than the median and because there are large outliers, the distribution is likely right-skewed with a single peak at the mode. b. How many families in the sample earned less than \$5,? Explain how you know. About 5 families (5% earned less than \$5, because that value is the median income. c. Based on the given information, can you determine how many families earned more than \$55,? Why or why not? Other than to say that is less than half, we don t have enough information to determine how many families earned more than \$55,. Chapter 6B Does it make sense? Decide whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. 9. For the 3 students who took the test, the high score was 8, the median was 7, and the low score was. Makes sense. Supposing half the students scored 7 or better this is entirely possible.. The mean gas mileage of the compact cars we tested was 3 miles per gallon, with a standard deviation of 5 gallons. Does not make sense. The standard deviation and mean should have the same units.

6 Comparing Variations. Consider the following data sets. a. Find the mean, median, and range for each of the two data sets. b. Give the five number summary and draw a boxplot for each of the two data sets. c. Find the standard deviation for each of the two data sets. d. Apply the range rule of thumb to estimate the standard deviation of each of the two data sets. How well does the rule work in each case? Briefly discuss why it does or does not work well. e. Based on all your results, compare and discuss the two data sets in terms of center and variation. 5. The table below gives the cost of living index for six East Coast cities and six West Coast cities (using the ACCRA index, where represents the average cost of living for all participating cities with a population over.5 million). East Coast Cities West Coast Cities Atlanta 98. Los Angeles 55.8 Baltimore 8.7 Portland 3. Boston 35. San Diego.8 Miami.5 San Francisco 8. New York City 6. San Jose 56. Washington, DC. Seattle.7 a. For the East Coast the mean, median, and range are 3.97, 3.5, and 7.8, respectively; and, for the West Coast they are 5.8, 5.3, and 69., respectively. b. For the East Coast the five-number summary is (98., 8.7, 3.5,., 6.), while for the West Coast, it is (3.,.7, 5.3, 56., 8.). The boxplots are then: East Coast West Coast c. The standard deviation is.86 for the East Coast, and 5.6 for the West Coast. d. For the East Coast approximate standard deviation is 7.8/ = 9.5, which is a far cry from the true value of.86 largely because of New York. On the West Coast, the approximate value is 69./ = 7.3, which is also fairly inaccurate. e. The cost of living is smaller, though more varied, on the East Coast. Understanding Variation. The following exercises give four data sets consisting of seven numbers. a. Make a histogram for each set. b. Give the five-number summary and draw a boxplot for each set. c. Compute the standard deviation for each set. d. Bases on your results, briefly explain how the standard deviation provides a useful single-number summary of the variation in these data sets.

7 . The following sets of numbers all have a mean of 6: {6,6,6,6,6,6,6},{5,5,6,6,6,7,7}, {5,5,5,6,7,7,7},{3,3,3,6,9,9,9} a b. The five number summaries for each of the sets are (in order): (6,6,6,6,6); (5,5,6,7,7); (5,5,6,7,7); (3,3,6,9,9). The boxplots are: Set Set Set 3 Set 6 8 c. The standard deviations for the sets are (in order):.,.86,., and 3.. d. Looking at part c, we can see that the variation increases with each successive set. 3. Portfolio Standard Deviation. The book Investments by Zvi Bodie, Alex Kane, and Alan Marcus claims that the returns for investment portfolios with a single stock have a standard deviation of.55, while the returns for portfolios with 3 stocks have a standard deviation of.35. Explain how the standard deviation measures the risk in these two types of portfolios. A lower standard deviation suggests more certainty in the expected return, and a lower risk.

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