# 3.E: Descriptive Statistics (Optional Exercises)

## 2.4: Measures of the Location of the Data

### Q 2.4.1

The median age for U.S. blacks currently is 30.9 years; for U.S. whites it is 42.3 years.

1. Based upon this information, give two reasons why the black median age could be lower than the white median age.
2. Does the lower median age for blacks necessarily mean that blacks die younger than whites? Why or why not?
3. How might it be possible for blacks and whites to die at approximately the same age, but for the median age for whites to be higher?

### Q 2.4.2

Six hundred adult Americans were asked by telephone poll, "What do you think constitutes a middle-class income?" The results are in Table. Also, include left endpoint, but not the right endpoint.

### S 2.8.12

For pianos, the cost of the piano is 0.4 standard deviations BELOW the mean. For guitars, the cost of the guitar is 0.25 standard deviations ABOVE the mean. For drums, the cost of the drum set is 1.0 standard deviations BELOW the mean. Of the three, the drums cost the lowest in comparison to the cost of other instruments of the same type. The guitar costs the most in comparison to the cost of other instruments of the same type.

### Q 2.8.13

An elementary school class ran one mile with a mean of 11 minutes and a standard deviation of three minutes. Rachel, a student in the class, ran one mile in eight minutes. A junior high school class ran one mile with a mean of nine minutes and a standard deviation of two minutes. Kenji, a student in the class, ran 1 mile in 8.5 minutes. A high school class ran one mile with a mean of seven minutes and a standard deviation of four minutes. Nedda, a student in the class, ran one mile in eight minutes.

1. Why is Kenji considered a better runner than Nedda, even though Nedda ran faster than he?
2. Who is the fastest runner with respect to his or her class? Explain why.

### Q 2.8.14

The most obese countries in the world have obesity rates that range from 11.4% to 74.6%. This data is summarized in Table 14.

Percent of Population Obese Number of Countries
11.4–20.45 29
20.45–29.45 13
29.45–38.45 4
38.45–47.45 0
47.45–56.45 2
56.45–65.45 1
65.45–74.45 0
74.45–83.45 1

What is the best estimate of the average obesity percentage for these countries? What is the standard deviation for the listed obesity rates? The United States has an average obesity rate of 33.9%. Is this rate above average or below? How “unusual” is the United States’ obesity rate compared to the average rate? Explain.

### S 2.8.14

• $$\bar{x} = 23.32$$
• Using the TI 83/84, we obtain a standard deviation of: $$s_{x} = 12.95$$.
• The obesity rate of the United States is 10.58% higher than the average obesity rate.
• Since the standard deviation is 12.95, we see that $$23.32 + 12.95 = 36.27$$ is the obesity percentage that is one standard deviation from the mean. The United States obesity rate is slightly less than one standard deviation from the mean. Therefore, we can assume that the United States, while 34% obese, does not have an unusually high percentage of obese people.

### Q 2.8.15

Table gives the percent of children under five considered to be underweight.

Percent of Underweight Children Number of Countries
16–21.45 23
21.45–26.9 4
26.9–32.35 9
32.35–37.8 7
37.8–43.25 6
43.25–48.7 1

What is the best estimate for the mean percentage of underweight children? What is the standard deviation? Which interval(s) could be considered unusual? Explain.