Question

consider the boxplot below.
16mins×
15 32 48 59 69
10 20 30 40 50 60 70
data

a. what quarter has the largest spread of data?
fourth quarter
second quarter
first quarter
third quarter
b. what is that spread?
c. find the inter quartile range (iqr):
d. which interval has the most data in it?
56 - 69
49 - 59
15 - 32
32 - 48
e. what value could represent the 79th percentile?
61
39
16
55

Explanation:

Step1: Recall box - plot properties

In a box - plot, the first quartile ($Q_1$) is 32, the second quartile ($Q_2$ or median) is 48, the third quartile ($Q_3$) is 59, the minimum is 15 and the maximum is 69.

Step2: Calculate spreads for each quarter

The spread of the first quarter is $Q_1 - \text{min}=32 - 15=17$. The spread of the second quarter is $Q_2 - Q_1=48 - 32 = 16$. The spread of the third quarter is $Q_3 - Q_2=59 - 48 = 11$. The spread of the fourth quarter is $\text{max}-Q_3=69 - 59 = 10$. So the first quarter has the largest spread.

Step3: Answer part b

The spread of the first quarter (the largest spread) is $32-15 = 17$.

Step4: Calculate the IQR

The inter - quartile range (IQR) is $Q_3 - Q_1=59 - 32=27$.

Step5: Determine the interval with most data

The box in the box - plot represents the middle 50% of the data. The interval of the box is from $Q_1 = 32$ to $Q_3 = 59$. Among the given intervals, the interval $32 - 48$ is part of the box and has the most data.

Step6: Estimate the 79th percentile

Since $Q_3$ is the 75th percentile and is 59, and the data is increasing, a value greater than 59 would be a reasonable estimate for the 79th percentile. Among the options, 61 is the value that could represent the 79th percentile.

Answer:

a. First Quarter
b. 17
c. 27
d. 32 - 48
e. 61