3.2 The average unemployment rate and the average inflation rate for the Carter years are calculated by adding up the four values of each for 1977 through 1980, and then dividing each sum by 4. Similarly, the averages for the Reagan years are calculated by adding up the eight values of each for 1981 through 1988, and then dividing each sum by 8. These calculations show that the Reagan years were characterized by a higher average rate of unemployment and a much lower average rate of inflation:

3.10 The mean is 14.7053 and the standard deviation is 7.8105:

    For a histogram, I divided the data into four intervals, and computed the relative frequencies and histogram heights (relative frequencies divided by interval width):

Here is the histogram:

For a box plot, we can find the quartiles by arranging the 30 observations in order:

The median is (12.91 + 12.31)/2 = 12.61; the first quartile is 8.92, the third quartile is 17.45, and the interquartile range is 17.45 - 8.92 = 8.53. The lower limit for outliers is 8.92 - 1.5(8.53) = -3.875. The upper limit for outliers is 17.45 + 1.5(8.53) = 30.245. Thus, the 30.57 and 34.04 values are outliers. Here is a modified box plot:

The mean plus or minus one standard deviation encompasses the range 14.7053 ± 7.8105 = 6.8948 to 22.5158; there are 20 observations (66.67 percent) in this range. The mean plus or minus two standard deviations encompasses the range 14.7053 ± 2(7.8105) = -0.9157 to 30.3263; there are 28 observation (93.33 percent) in this range. (Even though the histogram is not particularly bell-shaped, these percentages are roughly consistent with the rules of thumb that 68 percent are within one standard deviation and 95 percent are within two standard deviations.)

3.34 This is an example of Simpson's paradox. Because of its higher population growth rate, Costa Rica had far more young women than did Sweden. Because young women have relatively low death rates, the average female death rate was consequently lower in Costa Rica than in Sweden.

3.40 The exact answers are less important than whether a reasonable procedure is used. The histogram is Figure 2.13 seems to be roughly bell-shaped with a center at slightly less than 40 inches. Because the histogram is reasonably symmetrical, we might use 40 inches for the mean and median. For the standard deviation, we can use the rule of thumb that for a bell-shaped histogram, approximately 95 percent of the data are within two standard deviations. It appears that roughly that 95 percent of these data are between 35 and 45. If two standard deviations is equal to 5, then the standard deviation is 2.5 inches.

3.50 a. The median household is 93.8/2 = 46.9 million households in from either end; because this is among the 32.0 million people living in two-person households, the size of the median household is 2.
      b. The median individual is 264/2 = 132 million individuals in from either end. There are 23.0 million people in 1-person households, 2(30.2) = 60.4 million people in 2-person households, and 3(16.1) = 48.3 million people in 3-person households--giving a total of 131.7 million people in the first three categories. Thus the median individual lives in a four-person household.
      c. If we average the class size over the number of classes, a class with 100 students counts as 1 observation and a class with 1 student counts as 1 observation, giving an average class size of (100 + 1)/2 = 50.5. If we average the class size over the number of students, a class with 100 students counts as 100 observations and a class with 1 student counts as 1 observation, giving an average class size of (100(100) + 1)/101 = 99.0. (One hundred out of 101 students are in a class with 100 students.) From the professor's perspective, the average class has 50 students. For the student's perspective, the average class has 99 students, proving once again that it is better to be a professor than to be a student.