Question

a high school conducted a survey to find out the number of hours students spend on extracurricular activities per week. the results were recorded and organized into a back - to - back stem and leaf plot to compare the distributions between boys and girls. which of the following statements comparing the two distributions is true? (a) the distribution of hours for girls has a higher median than that of the boys. (b) the distribution of hours for boys is more symmetric than that of the girls. (c) the distribution of hours for boys has a larger range than that of the girls. (d) the distribution of hours for girls has a larger first quartile than that of the boys. (e) the distribution of hours for boys has the same iqr as that of the girls.

Explanation:

Step1: Count data - points

Boys have 15 data - points and girls have 15 data - points.

Step2: Find median position

For \(n = 15\) (odd number of data - points), the median position is \(\frac{n + 1}{2}=\frac{15+1}{2}=8\)th value when data is ordered.

Step3: Find boys' median

Ordering boys' data from the stem - and - leaf plot: \(05,05,06,06,08,08,09,10,11,12,13,13,14,16,19\). The 8th value is \(10\).

Step4: Find girls' median

Ordering girls' data from the stem - and - leaf plot: \(03,04,04,05,06,06,07,07,08,08,09,09,10,11,15\). The 8th value is \(07\). So option A is false.

Step5: Check symmetry

Boys' data is not symmetric as the left - hand side and right - hand side of the median do not have a similar spread. Girls' data is also not symmetric. But boys' data is more skewed than girls' data in some aspects, so option B is false.

Step6: Calculate range

Boys' range: \(19 - 5=14\). Girls' range: \(15 - 3 = 12\). So boys have a larger range, option C is true.

Step7: Calculate first quartile (\(Q_1\))

For \(n = 15\), the position of \(Q_1\) is \(\frac{n + 1}{4}=\frac{15 + 1}{4}=4\)th value when data is ordered. Boys' \(Q_1\): 6. Girls' \(Q_1\): 4. So option D is false.

Step8: Calculate IQR

For boys, position of \(Q_3\) is \(3\times\frac{n + 1}{4}=12\)th value. Boys' \(Q_3\) is 13, \(IQR_{boys}=13 - 6 = 7\). For girls, \(Q_3\) is 9, \(IQR_{girls}=9 - 4=5\). So option E is false.

Answer:

C. The distribution of hours for boys has a larger range than that of the girls.