12.9 Median income. You read that the median income of U.S. households in 2013 was $51,939. Explain in plain language what “the median income’’ is.

12.10 What’s the average? The Census Bureau website gives several choices for ‘‘average income’’ in its historical income data. In 2013, the median income of American households was $51,939. The mean household income was $72,641. The median income of families was $63,815, and the mean family income was $84,687. The Census Bureau says, “Households consist of all people who occupy a housing unit. The term ‘family’ refers to a group of two or more people related by birth, marriage, or adoption who reside together.’’ Explain carefully why mean incomes are higher than median incomes and why family incomes are higher than household incomes.

12.11 Rich magazine readers. Echo Media reports that the average income for readers of the magazine WatchTime (a magazine for people interested in fine watches) is $298,400. Is the median wealth of these readers greater or less than $298,400? Why?

12.12 College tuition. Figure 11.7 (page 255) is a stemplot of the tuition charged by 116 colleges in Illinois. The stems are thousands of dollars and the leaves are hundreds of dollars. For example, the highest tuition is $38,600 and appears as leaf 6 on stem 38.

12.13 Where are the young more likely to live? Figure 11.11 (page 259) is a stemplot of the percentage of residents aged 18 to 34 in each of the 50 states. The stems are whole percents and the leaves are tenths of a percent.

12.14 Gas mileage. Table 11.2 (page 261) gives the highway gas mileages for some model year 2015 mid-sized cars.

12.15 Yankee money. Table 11.4 (page 262) gives the salaries of the New York Yankees baseball team. What shape do you expect the distribution to have? Do you expect the mean salary to be close to the median, clearly higher, or clearly lower? Verify your choices by making a graph and calculating the mean and median.

12.16 The richest 5%. The distribution of individual incomes in the United States is strongly skewed to the right. In 2013, the mean and median incomes of the top 5% of Americans were $196,723 and $322,343. Which of these numbers is the mean and which is the median? Explain your reasoning.

12.17 How many calories does a hot dog have? Consumer Reports magazine presented the following data on the number of calories in a hot dog for each of 17 brands of meat hot dogs:

Make a stemplot [if you did not already do so in Exercise 11.19 (page 263)], and find the five-number summary. The stemplot shows important facts about the distribution that the numerical summary does not tell us. What are these facts?

12.18 Returns on common stocks. Example 5 informs us that financial theory uses the mean and standard deviation to describe the returns on investments. Figure 11.13 (page 260) is a histogram of the returns of all New York Stock Exchange common stocks in one year. Are the mean and standard deviation suitable as a brief description of this distribution? Why?

12.19 Minority students in engineering. Figure 11.12 (page 259) is a histogram of the number of minority students (black, Hispanic, Native American) who earned doctoral degrees in engineering from each of 152 universities in the years 2000 through 2002. The classes for Figure 11.12 are 1–5, 6–10, and so on.

12.20 The statistics of writing style. Here are data on the percentages of words of 1 to 15 letters used in articles in Popular Science magazine. Exercise 11.16 (page 263) asked you to make a histogram of these data.

Find the five-number summary of the distribution of word lengths from this table.

12.21 Immigrants in the eastern states. Here are the number of legal immigrants (in thousands) who settled in each state east of the Mississippi River in 2013:

Make a graph of the distribution. Describe its overall shape and any outliers. Then choose and calculate a suitable numerical summary.

12.22 Immigrants in the eastern states. New York and Florida are high outliers in the distribution of the previous exercise. Find the mean and the median for these data with and without New York and Florida. Which measure changes more when we omit the outliers?

12.23 State SAT scores. Figure 12.9 is a histogram of the average scores on the SAT Mathematics exam for college-bound senior students in the 50 states and the District of Columbia in 2014. The distinctive overall shape of this distribution implies that a single measure of center such as the mean or the median is of little value in describing the distribution. Explain why this is true.

12.24 Highly paid athletes. A news article reported that of the 411 players on National Basketball Association rosters in February 1998, only 139 “made more than the league average salary’’ of $2.36 million. Was $2.36 million the mean or median salary for NBA players? How do you know?

12.25 Mean or median? Which measure of center, the mean or the median, should you use in each of the following situations? Why?

12.26 Mean or median? You are planning a party and want to know how many cans of soda to buy. A genie offers to tell you either the mean number of cans guests will drink or the median number of cans. Which measure of center should you ask for? Why? To make your answer concrete, suppose there will be 30 guests and the genie will tell you either cans or cans. Which of these two measures would best help you determine how many cans you should have on hand?

12.27 State SAT scores. We want to compare the distributions of average SAT Math and Writing scores for the states and the District of Columbia. We enter these data into a computer with the names SATM for Math scores and SATW for Writing scores. At the bottom of the page is output from the statistical software package Minitab. (Other software produces similar output. Some software uses rules for finding the quartiles that differ slightly from ours. So software may not give exactly the answer you would get by hand.)

Use this output to make boxplots of SAT Math and Writing scores for the states. Briefly compare the two distributions in words.

12.28 Do SUVs waste gas? Table 11.2 (page 261) gives the highway fuel consumption (in miles per gallon) for 31 model year 2015 mid-sized cars. You found the five-number summary for these data in Exercise 12.14. Here are the highway gas mileages for 26 four-wheel-drive model year 2015 sport utility vehicles:

12.29 How many calories in a hot dog? Some people worry about how many calories they consume. Consumer Reports magazine, in a story on hot dogs, measured the calories in 20 brands of beef hot dogs, 17 brands of meat hot dogs, and 17 brands of poultry hot dogs. Here is computer output describing the beef hot dogs,

Mean = 156.8 Standard deviation = 22.64 Min = 111 Max = 190 N = 20 Median = 152.5  Quartiles = 140, 178.5

the meat hot dogs,

Mean = 158.7 Standard deviation = 25.24 Min = 107 Max = 195 N = 17 Median = 153  Quartiles = 139, 179

and the poultry hot dogs,

Mean = 122.5 Standard deviation = 25.48  Min = 87 Max = 170 N = 17 Median = 129 Quartiles = 102, 143

(Some software uses rules for finding the quartiles that differ slightly from ours. So software may not give exactly the answer you would get by hand.) Use this information to make boxplots of the calorie counts for the three types of hot dogs. Write a brief comparison of the distributions. Will eating poultry hot dogs usually lower your calorie consumption compared with eating beef or meat hot dogs? Explain.

12.30 Finding the standard deviation. The level of various substances in the blood influences our health. Here are measurements of the level of phosphate in the blood of a patient, in milligrams of phosphate per deciliter of blood, made on six consecutive visits to a clinic:

A graph of only six observations gives little information, so we proceed to compute the mean and standard deviation.

12.31 What s measures. Use a calculator to find the mean and standard deviation of these two sets of numbers:

Which data set is more variable?

12.32 What s measures. Add 2 to each of the numbers in data set (a) in the previous exercise. The data are now 6 2 3 6 5 8.

12.33 Cars and SUVs. Use the mean and standard deviation to compare the gas mileages of mid-size cars (Table 11.2, page 261) and SUVs (Exercise 12.28). Do these numbers catch the main points of your more detailed comparison in Exercise 12.28?

12.34 A contest. This is a standard deviation contest. You must choose four numbers from the whole numbers 0 to 9, with repeats allowed.

12.35 and s are not enough. The mean and standard deviation s measure center and variability but are not a complete description of a distribution. Data sets with different shapes can have the same mean and standard deviation. To demonstrate this fact, use your calculator to find and s for these two small data sets. Then make a stemplot of each and comment on the shape of each distribution.

12.36 Raising pay. A school system employs teachers at salaries between $40,000 and $70,000. The teachers’ union and the school board are negotiating the form of next year’s increase in the salary schedule. Suppose that every teacher is given a flat $3000 raise.

12.37 Raising pay. Suppose that the teachers in the previous exercise each receive a 5% raise. The amount of the raise will vary from $2000 to $3500, depending on present salary. Will a 5% across-the-board raise increase the variability of the distribution as measured by the distance between the quartiles? Do you think it will increase the standard deviation? Explain your reasoning.

12.38 Making colleges look good. Colleges announce an “average’’ SAT score for their entering freshmen. Usually the college would like this “average’’ to be as high as possible. A New York Times article noted, “Private colleges that buy lots of top students with merit scholarships prefer the mean, while open-enrollment public institutions like medians.’’ Use what you know about the behavior of means and medians to explain these preferences.

12.39 What graph to draw? We now understand three kinds of graphs to display distributions of quantitative variables: histograms, stemplots, and boxplots. Give an example (just words, no data) of a situation in which you would prefer that graphing method.