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

a. find the data set’s first, second, and third quartiles.
q₁ = □
q₂ = □
q₃ = □
(type integers or decimals. do not round.)

Explanation:

Step 1: Sort the data

First, we sort the given data set in ascending order:
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 position of quartiles

For a data set with \( n = 25 \) observations, the position of the second quartile (\( Q_2 \), which is the median) is given by \( \frac{n + 1}{2}=\frac{25+ 1}{2}=13 \)th term.

The position of the first quartile (\( Q_1 \)) is given by \( \frac{n + 1}{4}=\frac{25 + 1}{4}=6.5 \)th term.

The position of the third quartile (\( Q_3 \)) is given by \( \frac{3(n + 1)}{4}=\frac{3\times(25 + 1)}{4}=19.5 \)th term.

Step 3: Calculate \( Q_1 \)

The 6th term is \( 15.00 \) and the 7th term is \( 15.20 \). For the 6.5th term (average of 6th and 7th term):
\( Q_1=\frac{15.00 + 15.20}{2}=\frac{30.20}{2}=15.10 \)

Step 4: Calculate \( Q_2 \)

The 13th term in the sorted data is \( 15.85 \), so \( Q_2 = 15.85 \)

Step 5: Calculate \( Q_3 \)

The 19th term is \( 17.60 \) and the 20th term is \( 17.70 \). For the 19.5th term (average of 19th and 20th term):
\( Q_3=\frac{17.60+17.70}{2}=\frac{35.30}{2}=17.65 \)

Answer:

\( Q_1 = 15.10 \)
\( Q_2 = 15.85 \)
\( Q_3 = 17.65 \)