Question

question four
a survey was given to a random sample of 20 sophomore college students. they were asked, “how many textbooks do you own?” their responses were: 0, 0, 2, 5, 8, 8, 8, 9, 9, 10, 10, 10, 11, 12, 12, 12, 14, 15, 20, and 25.
the observations are in order from smallest to largest.
(a) determine the quartiles.
q1 = 8.0
median(q2)=10.0
q3 = 13.0
(b) compute the iqr.
iqr = q3 - q1 = 5.0
(c) write the five - number summary.
(min, q1, median, q3, max)=(0, 8.0, 10.0, 13.0, 25)
(d) draw a box plot

Explanation:

Step1: Find position of Q1

For a sample of size $n = 20$, the position of $Q1$ is $0.25\times(n + 1)=0.25\times(20 + 1)=5.25$. The 5th value is 8 and the 6th value is 8. So $Q1=\frac{8 + 8}{2}=8$.

Step2: Find position of Q2 (Median)

The position of $Q2$ is $0.5\times(n + 1)=0.5\times(20+1)=10.5$. The 10th value is 10 and the 11th value is 10. So $Q2=\frac{10 + 10}{2}=10$.

Step3: Find position of Q3

The position of $Q3$ is $0.75\times(n + 1)=0.75\times(20 + 1)=15.75$. The 15th value is 12 and the 16th value is 14. So $Q3=\frac{12+14}{2}=13$.

Step4: Compute IQR

$IQR=Q3 - Q1=13 - 8 = 5$.

Step5: Write five - number summary

The five - number summary is (Min, Q1, Median, Q3, Max). Min = 0, Q1 = 8, Median=10, Q3 = 13, Max = 25. So it is $(0,8,10,13,25)$.

Step6: Drawing box - plot (description)

Draw a number line that includes the range from 0 to 25. Draw a box from Q1 = 8 to Q3 = 13. Draw a vertical line inside the box at the median value of 10. Draw whiskers from the box to the minimum value 0 and the maximum value 25.

Answer:

(a) $Q1 = 8$, $Q2=10$, $Q3 = 13$
(b) $IQR = 5$
(c) $(0,8,10,13,25)$
(d) Draw a box from 8 to 13 with a vertical line at 10 and whiskers to 0 and 25 on a number - line from 0 to 25.