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)
outliers(using iqr rule):
lower fence = 2.0, upper fence = 18.0, outliers: 0, 0, 20, 25
(d) draw a box plot

Explanation:

Step1: Calculate quartiles

For a sample of size $n = 20$ (even - numbered data set). The median (Q2) is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+ 1)$th ordered values. $\frac{n}{2}=10$, so Q2 is the average of the 10th and 11th values, which are 10 and 10, so $Q2=\frac{10 + 10}{2}=10$. The lower half of the data (first 10 values) is used to find Q1. Since $n_1 = 10$ (even), Q1 is the average of the 5th and 6th values, which are 8 and 8, so $Q1=\frac{8+8}{2}=8$. The upper - half of the data (last 10 values) is used to find Q3. Since $n_2 = 10$ (even), Q3 is the average of the 5th and 6th values of the upper - half, which are 12 and 14, so $Q3=\frac{12 + 14}{2}=13$.

Step2: Compute the IQR

The inter - quartile range $IQR=Q3 - Q1$. Substituting the values of Q1 and Q3, we get $IQR=13 - 8=5$.

Step3: Write the five - number summary

The five - number summary consists of the minimum value, Q1, Q2 (median), Q3, and the maximum value. The minimum value in the data set is 0 and the maximum value is 25. So the five - number summary is $(0,8,10,13,25)$.

Step4: Identify outliers

The lower fence for outliers is $Q1-1.5\times IQR=8-1.5\times5=8 - 7.5 = 0.5$. The upper fence for outliers is $Q3 + 1.5\times IQR=13+1.5\times5=13 + 7.5=20.5$. Values less than 0.5 or greater than 20.5 are outliers. So the outliers are 0, 20, 25.

Step5: Draw a box plot (description)

On a number line, mark the five - number summary values. Draw a box from Q1 to Q3 with a line inside the box at Q2. Draw whiskers from the box to the non - outlier minimum and maximum values (excluding outliers in the whisker length). Mark the outliers (0, 20, 25) as individual points.

Answer:

(a) $Q1 = 8$, $Q2=10$, $Q3 = 13$
(b) $IQR = 5$
(c) $(0,8,10,13,25)$
(d) Draw a box with ends at Q1 and Q3, a line in the box at Q2, whiskers to non - outlier min and max, and mark outliers 0, 20, 25 as individual points.