Question

a student takes ten exams during a semester and receives the following grades: 90, 85, 97, 76, 89, 58, 82, 102, 70, and 67. find the five - number summary used in a boxplot.
a. 58, 70, 83.5, 90, 102
b. 102, 90, 83.5, 70, 58
c. 58, 70.75, 83.5, 92.75, 102
d. 102, 92.75, 83.5, 70.75, 58
please select the best answer from the choices provided

Explanation:

Step1: Sort the data

Sort the grades: \(58,67,70,76,82,85,89,90,97,102\)

Step2: Find the minimum

The minimum value is \(58\)

Step3: Find the first quartile (\(Q_1\))

Since \(n = 10\), the position of \(Q_1\) is \(\frac{n + 1}{4}=\frac{10+ 1}{4}=2.75\)
\(Q_1=70+(76 - 70)\times0.75=70 + 4.5=74.5\) (Wait, no, correct formula for \(Q_1\) when \(n\) even: split the data into lower half \(\{58,67,70,76,82\}\), \(Q_1=\frac{70 + 76}{2}=73\) (Wrong approach above). Correct way: for \(n = 10\) (even), lower half is first \(5\) numbers. \(Q_1=\frac{70+76}{2} = 73\) (No, another way: formula \(Q_1\) for \(n\) data points: if \(n\) even, \(Q_1\) is median of first \(\frac{n}{2}\) data. First \(5\) data \(58,67,70,76,82\), median \(70\) (Wait, no, median of a set with odd number of elements is the middle one. For \(5\) elements, median is the \(3^{rd}\) element. So \(Q_1 = 70\) (Wait, no, standard method: sort data \(x_1\leq x_2\leq\cdots\leq x_{10}\). \(Q_1=x_{3}\) when \(n = 10\) (using \(Q_1\) formula \(i=\frac{n + 1}{4}\), \(i = 2.75\), linear - interpolation: \(x_2+(x_3 - x_2)\times0.75\). \(x_2 = 67\), \(x_3=70\), \(Q_1=67+(70 - 67)\times0.75=67 + 2.25=69.25\) (No, wrong. Correct: \(n = 10\), lower half \(\{58,67,70,76,82\}\), \(Q_1\) is median of lower half. For \(n_1=5\) (lower half), median is \(70\).

Step4: Find the median (\(Q_2\))

Since \(n = 10\) (even), \(Q_2=\frac{x_5+x_6}{2}=\frac{82 + 85}{2}=83.5\)

Step5: Find the third quartile (\(Q_3\))

Upper half \(\{89,90,97,102\}\), \(Q_3=\frac{90+97}{2}=93.5\) (Wait, no, for \(n = 10\), upper half is last \(5\) numbers \(\{85,89,90,97,102\}\), median of upper half (for \(n_2 = 5\)) is \(90\) (Wait, no, standard method: \(i=\frac{3(n + 1)}{4}=\frac{3\times(10 + 1)}{4}=8.25\). \(x_8 = 90\), \(x_9=97\), \(Q_3=90+(97 - 90)\times0.25=90+1.75 = 91.75\) (No, correct: upper half of sorted data \(\{85,89,90,97,102\}\), median (third quartile) is \(90\)

Step6: Find the maximum

The maximum value is \(102\)

The five - number summary is \(58,70,83.5,90,102\)

Answer:

A. \(58,70,83.5,90,102\)