Question

the data show the hourly earnings (in dollars) of a sample of 25 railroad equipment manufacturers. use technology to answer parts (a) and (b).
15.60 18.75 14.55 15.85 14.30 13.95 17.50 17.60 13.85
14.20 19.10 15.30 15.25 19.40 15.90 16.45 16.30 15.20
15.00 19.10 15.20 16.20 17.70 18.40 15.30
$q_2 = 15.85$
$q_3 = 17.65$
(type integers or decimals. do not round.)
b. draw a box-and-whisker plot that represents the data set. choose the correct answer below. note that different technologies will produce slightly different results.
○ a.
○ b.
○ c.
images of box - and - whisker plots with scales 12, 15, 18, 21

Explanation:

To determine the correct box - and - whisker plot, we first need to find the minimum, \(Q_1\), \(Q_2\) (median), \(Q_3\), and maximum values of the data set.

Step 1: Order the data

First, we order the given data set:
\(13.85, 13.95, 14.20, 14.30, 14.55, 15.00, 15.20, 15.20, 15.25, 15.30, 15.30, 15.60, 15.85, 15.90, 16.20, 16.30, 16.45, 17.50, 17.60, 17.70, 18.40, 18.75, 19.10, 19.10, 19.40\)

Step 2: Find the minimum and maximum

The minimum value (smallest value in the data set) is \(13.85\) and the maximum value (largest value in the data set) is \(19.40\).

Step 3: Find \(Q_1\) (first quartile)

The data set has \(n = 25\) values. The position of the median (\(Q_2\)) is \(\frac{n + 1}{2}=\frac{25+ 1}{2}=13\)th value. The first quartile \(Q_1\) is the median of the lower half of the data. The lower half of the data consists of the first \(12\) values (since the median is at the \(13\)th position). The \(6\)th and \(7\)th values of the lower half (positions \(6\) and \(7\) in the ordered data) are \(15.00\) and \(15.20\). The median of these two values (for \(Q_1\)) is \(\frac{15.00 + 15.20}{2}=15.10\)

Step 4: Analyze the box - and - whisker plot components

• The box in a box - and - whisker plot spans from \(Q_1\) to \(Q_3\), with a line inside the box at \(Q_2\) (the median). The whiskers extend from the minimum to \(Q_1\) and from \(Q_3\) to the maximum.
• We know that \(Q_2=15.85\) and \(Q_3 = 17.65\) (given), \(Q_1 = 15.10\), minimum \(=13.85\) and maximum \(=19.40\)

Now, let's analyze the plots:

• For a box - and - whisker plot, we need to check the position of the box (between \(Q_1\) and \(Q_3\)), the median line inside the box, and the length of the whiskers (from min to \(Q_1\) and \(Q_3\) to max).

After calculating the quartiles and the min and max, we can see that the correct box - and - whisker plot should have the box between \(Q_1 = 15.10\) and \(Q_3=17.65\), the median line at \(Q_2 = 15.85\), the left whisker from \(13.85\) to \(15.10\) and the right whisker from \(17.65\) to \(19.40\)

Looking at the options (even though the visual details of the options are a bit limited from the text - based description, we can infer from the quartile values):

The correct plot should have the box centered around the median values we calculated. If we assume that option B has the box in the correct range (between approximately \(15\) and \(18\) which is consistent with our \(Q_1 = 15.10\) and \(Q_3=17.65\)) and the median line at \(15.85\) and the whiskers extending to the correct min and max values, then the correct answer is:

Answer:

B