To find the data set's first, second, and third quartiles, we'll proceed in the following steps:
1. Sort the Data:
The first step in finding the quartiles is to sort the data in ascending order. The sorted hourly earnings data is as follows:
13.80, 13.85, 14.20, 14.35, 14.60, 15.10, 15.15, 15.25, 15.25, 15.30, 15.40, 15.55, 15.80, 16.00, 16.25, 16.35, 16.50, 17.45, 17.50, 17.75, 18.40, 18.75, 19.10, 19.15, 19.45
2. Calculate the First Quartile (Q1):
The first quartile (Q1) is the median of the lower half of the data set. For a data set of 25 values, Q1 can be found at the 25th percentile.
- For 25 data points, the position for Q1 is [tex]\( 0.25 \times (25 + 1) = 0.25 \times 26 = 6.5 \)[/tex].
- Q1 will be the value between the 6th and 7th values in the sorted data.
- In our data, the 6th value is 15.10 and the 7th value is 15.15.
- Therefore, Q1 is exactly the value of the 6.5th position, which is 15.15.
3. Calculate the Second Quartile (Q2):
The second quartile (Q2) is the median of the entire data set.
- For 25 data points, Q2 is at the [tex]\( 0.50 \times (25 + 1) = 0.50 \times 26 = 13 \)[/tex].
- Therefore, Q2 is the 13th value in the sorted data.
- The 13th value in our data is 15.80.
4. Calculate the Third Quartile (Q3):
The third quartile (Q3) is the median of the upper half of the data set. For a data set of 25 values, Q3 can be found at the 75th percentile.
- For 25 data points, the position for Q3 is [tex]\( 0.75 \times (25 + 1) = 0.75 \times 26 = 19.5 \)[/tex].
- Q3 will be the value between the 19th and 20th values in the sorted data.
- In our data, the 19th value is 17.50 and the 20th value is 17.75.
- Therefore, Q3 is exactly the value of the 19.5th position, which is 17.50.
So, the quartiles are:
[tex]\[
\begin{array}{l}
Q _1 = 15.15 \\
Q _2 = 15.80 \\
Q _3 = 17.50
\end{array}
\][/tex]
