Question

the accompanying data represent the wait - time (in minutes) for a random sample of forty visitors to an amusement park ride. complete parts (a) and (b). (a) determine and interpret the quartiles. by the quartiles, about 25% of the wait times are q1 = 6 minute(s) or less, and about 75% of the wait times exceed q1 minute(s); about 50% of the wait times are q2 = 10.5 minute(s) or less and about 50% of the wait times exceed q2 minute(s); about 75% of the wait times are q3 = 19.75 minute(s) or less, and about 25% of the wait times exceed q3 minute(s). (type integers or decimals. do not round.)

Explanation:

Step1: Sort the data

First, sort the 40 - data set in ascending order: 1, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 10, 11, 14, 14, 15, 15, 15, 16, 16, 17, 19, 20, 22, 23, 26, 28, 31, 31, 43, 44, 62.

Step2: Calculate the position of Q1

The formula for the position of the first - quartile $Q_1$ for a data set of size $n$ is $i_1=\frac{1}{4}(n + 1)$. Here $n = 40$, so $i_1=\frac{1}{4}(40+1)=10.25$. The value of $Q_1$ is the value at the 10th position plus 0.25 times the difference between the 11th and 10th values. The 10th value is 6 and the 11th value is 6, so $Q_1=6+0.25\times(6 - 6)=6$.

Step3: Calculate the position of Q2

The formula for the position of the second - quartile (median) $Q_2$ is $i_2=\frac{1}{2}(n + 1)$. For $n = 40$, $i_2=\frac{1}{2}(40 + 1)=20.5$. The value of $Q_2$ is the average of the 20th and 21st values. The 20th value is 10 and the 21st value is 11, so $Q_2=\frac{10 + 11}{2}=10.5$.

Step4: Calculate the position of Q3

The formula for the position of the third - quartile $Q_3$ is $i_3=\frac{3}{4}(n + 1)$. For $n = 40$, $i_3=\frac{3}{4}(40+1)=30.75$. The value of $Q_3$ is the value at the 30th position plus 0.75 times the difference between the 31st and 30th values. The 30th value is 19 and the 31st value is 20, so $Q_3=19+0.75\times(20 - 19)=19.75$.

Answer:

$Q_1 = 6$, $Q_2=10.5$, $Q_3 = 19.75$