The table lists the heights (in centimeters) of preschool girls and boys on a playground.

\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Heights of Preschool Boys \\
(centimeters)
\end{tabular} & \begin{tabular}{c}
Heights of Preschool Girls \\
(centimeters)
\end{tabular} \\
\hline 105.1 & 104.8 \\
\hline 101.3 & 87 \\
\hline 86.7 & 95 \\
\hline 93.8 & 92.1 \\
\hline 92.4 & 100 \\
\hline 85.2 & 90.3 \\
\hline 99.6 & 98.6 \\
\hline 97.5 & 101.7 \\
\hline 102.9 & 89.4 \\
\hline 107 & 92.1 \\
\hline
\end{tabular}

Based on the table, which of the following is true?

A. The difference of the medians is about one-half the interquartile range of either data set.
B. The difference of the medians is about one-fourth the interquartile range of either data set.
C. The medians cannot be compared based on their interquartile ranges because the interquartile ranges are 1 centimeter apart.
D. The medians cannot be compared based on their interquartile ranges because the interquartile ranges are 3 centimeters apart.

Question

25-07-2024
Mathematics

Answered

The table lists the heights (in centimeters) of preschool girls and boys on a playground.

\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Heights of Preschool Boys \\
(centimeters)
\end{tabular} & \begin{tabular}{c}
Heights of Preschool Girls \\
(centimeters)
\end{tabular} \\
\hline 105.1 & 104.8 \\
\hline 101.3 & 87 \\
\hline 86.7 & 95 \\
\hline 93.8 & 92.1 \\
\hline 92.4 & 100 \\
\hline 85.2 & 90.3 \\
\hline 99.6 & 98.6 \\
\hline 97.5 & 101.7 \\
\hline 102.9 & 89.4 \\
\hline 107 & 92.1 \\
\hline
\end{tabular}

Based on the table, which of the following is true?

A. The difference of the medians is about one-half the interquartile range of either data set.
B. The difference of the medians is about one-fourth the interquartile range of either data set.
C. The medians cannot be compared based on their interquartile ranges because the interquartile ranges are 1 centimeter apart.
D. The medians cannot be compared based on their interquartile ranges because the interquartile ranges are 3 centimeters apart.

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-07-25T11:47:19-04:00

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.