Answer
American Women: IQR = 3 inches
American Men: IQR = 4 inches
Work Step by Step
Below are the steps to determine the interquartile ranges (IQR) for each group. Recall that for a data set with 50 observations (an even number), a common method is to split the ordered data into two halves (observations 1–25 and 26–50), then the first quartile (Q₁) is the median of the lower half (the 13th observation) and the third quartile (Q₃) is the median of the upper half (the 13th observation of that half, i.e. the 38th overall). The IQR is Q₃ – Q₁.
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### American Women in Their 20s
**Frequency distribution (from earlier work):**
| Height (in) | Frequency | Cumulative Frequency |
|-------------|-----------|----------------------|
| 59 | 2 | 2 |
| 60 | 4 | 6 |
| 61 | 1 | 7 |
| 62 | 4 | 11 |
| 63 | 7 | 18 |
| 64 | 9 | 27 |
| 65 | 10 | 37 |
| 66 | 6 | 43 |
| 67 | 3 | 46 |
| 68 | 1 | 47 |
| 69 | 1 | 48 |
| 70 | 1 | 49 |
| 72 | 1 | 50 |
- **Lower Half (Observations 1–25):**
The 13th observation falls in the 63‐inch group (since cumulative frequency reaches 11 at 62 inches and 18 at 63 inches).
**Q₁ = 63 inches.**
- **Upper Half (Observations 26–50):**
The 38th observation is the 13th observation in this half. Since cumulative frequency is 27 at 64 inches and 37 at 65 inches, the 38th observation is the first observation in the next group, 66 inches.
**Q₃ = 66 inches.**
- **Interquartile Range:**
\[
\text{IQR} = Q₃ - Q₁ = 66 - 63 = 3 \text{ inches.}
\]
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### American Men in Their 20s
**Frequency distribution (from earlier work):**
| Height (in) | Frequency | Cumulative Frequency |
|-------------|-----------|----------------------|
| 60 | 2 | 2 |
| 61 | 1 | 3 |
| 62 | 1 | 4 |
| 64 | 1 | 5 |
| 65 | 3 | 8 |
| 66 | 2 | 10 |
| 67 | 1 | 11 |
| 68 | 4 | 15 |
| 69 | 6 | 21 |
| 70 | 7 | 28 |
| 71 | 6 | 34 |
| 72 | 7 | 41 |
| 73 | 4 | 45 |
| 74 | 1 | 46 |
| 75 | 2 | 48 |
| 76 | 1 | 49 |
| 77 | 1 | 50 |
- **Lower Half (Observations 1–25):**
The 13th observation falls in the 68‐inch group (cumulative frequency is 11 at 67 inches and 15 at 68 inches).
**Q₁ = 68 inches.**
- **Upper Half (Observations 26–50):**
The 38th observation falls in the 72‐inch group (cumulative frequency is 34 at 71 inches and 41 at 72 inches).
**Q₃ = 72 inches.**
- **Interquartile Range:**
\[
\text{IQR} = Q₃ - Q₁ = 72 - 68 = 4 \text{ inches.}
\]
