To compare the medians of the heights of the preschool boys and girls based on their interquartile ranges (IQRs), let's break down the given information systematically.
### Calculating the Medians
1. Heights of Preschool Boys:
Given heights are: 105.1, 101.3, 86.7, 93.8, 92.4, 85.2, 99.6, 97.5, 102.9, 107
- The median height of boys is 98.55 cm.
2. Heights of Preschool Girls:
Given heights are: 104.8, 87, 95, 92.1, 100, 90.3, 98.6, 101.7, 89.4, 92.1
- The median height of girls is 93.55 cm.
### Difference of the Medians
The difference in medians between the boys and girls:
[tex]\[ \text{Median Difference} = 98.55 - 93.55 = 5.0 \text{ cm} \][/tex]
### Calculating the Interquartile Ranges (IQRs)
1. Interquartile Range for Boys:
- Q1 (25th percentile) = 92.4 cm
- Q3 (75th percentile) = 102.15 cm
- IQR Boys: [tex]\( \text{IQR}_{\text{Boys}} = Q3 - Q1 = 102.15 - 92.4 = 9.75 \text{ cm} \)[/tex]
2. Interquartile Range for Girls:
- Q1 (25th percentile) = 90.3 cm
- Q3 (75th percentile) = 99.2 cm
- IQR Girls: [tex]\( \text{IQR}_{\text{Girls}} = Q3 - Q1 = 99.2 - 90.3 = 8.9 \text{ cm} \)[/tex]
### Comparing the Medians with IQR
- Half the IQR Boys: [tex]\( \frac{9.75}{2} = 4.875 \)[/tex] cm
- One-fourth the IQR Boys: [tex]\( \frac{9.75}{4} = 2.4375 \)[/tex] cm
- Half the IQR Girls: [tex]\( \frac{8.9}{2} = 4.45 \)[/tex] cm
- One-fourth the IQR Girls: [tex]\( \frac{8.9}{4} = 2.225 \)[/tex] cm
### Conclusion
The difference of the medians (5.0 cm) is closest to being about one-half the interquartile range of either data set (Boys: 4.875 cm and Girls: 4.45 cm).
Therefore, the statement that is true is:
A. The difference of the medians is about one-half the interquartile range of either data set.
