Question

the data show the distance (in miles) from an airport of a sample of 22 inbound and outbound airplanes. use technology to answer parts (a) and (b). a. find the data sets first, second, and third quartiles. b. draw a box - and - whisker plot that represents the data set. 4.8 3.2 3.8 4.7 3.9 3.8 2.9 3.9 2.4 2.6 5.2 2.1 4.3 2.9 3.5 4.9 1.8 2.1 4.6 4.6 4.1 2.1 a. find the three quartiles. q1 = □ q2 = □ q3 = □ (type integers or decimals. do not round.)

Explanation:

Step1: Sort the data set

1.8, 2.1, 2.1, 2.1, 2.4, 2.6, 2.9, 2.9, 3.2, 3.5, 3.8, 3.8, 3.9, 3.9, 4.1, 4.3, 4.6, 4.6, 4.7, 4.8, 4.9, 5.2

Step2: Calculate the position of Q2 (median)

Since \(n = 22\) (even), \(Q2=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}\). \(\frac{n}{2}=11\), \(\frac{n}{2}+1 = 12\). So \(Q2=\frac{3.8 + 3.8}{2}=3.8\)

Step3: Calculate the position of Q1

The lower - half of the data set has \(n_1 = 11\) values. Since \(n_1\) is odd, \(Q1=x_{\frac{n_1 + 1}{2}}\), \(\frac{n_1+1}{2}=6\). So \(Q1 = 2.6\)

Step4: Calculate the position of Q3

The upper - half of the data set has \(n_2=11\) values. Since \(n_2\) is odd, \(Q3=x_{\frac{n_2 + 1}{2}}\), \(\frac{n_2+1}{2}=6\) in the upper - half. So \(Q3 = 4.6\)

Answer:

\(Q1 = 2.6\)
\(Q2 = 3.8\)
\(Q3 = 4.6\)

To draw the box - and - whisker plot:

1. Draw a number line that includes the range of the data (from 1.8 to 5.2).
2. Mark \(Q1 = 2.6\), \(Q2=3.8\) and \(Q3 = 4.6\) on the number line.
3. Draw a box from \(Q1\) to \(Q3\) with a vertical line at \(Q2\) inside the box.
4. The lower whisker extends from \(Q1\) to the minimum value (1.8) and the upper whisker extends from \(Q3\) to the maximum value (5.2).