To determine the amount that separates the lowest 75% of the data from the highest 25% of the data, we need to calculate the 75th percentile of the monthly revenue dataset. In other words, we need to find the value below which 75% of the revenue data falls.
Here is a step-by-step approach to calculate the 75th percentile:
1. List the data in ascending order:
The given monthly revenue data is:
[tex]\[
27,550, 19,700, 35,850, 41,830, 43,520, 24,900, 28,590, 25,170, 31,860, 17,440, 54,470, 70,280, 45,270, 35,850
\][/tex]
Sorting this in ascending order:
[tex]\[
17,440, 19,700, 24,900, 25,170, 27,550, 28,590, 31,860, 35,850, 35,850, 41,830, 43,520, 45,270, 54,470, 70,280
\][/tex]
2. Determine the position of the 75th percentile:
The position [tex]\( P \)[/tex] of the 75th percentile in a sorted list of [tex]\( n \)[/tex] values can be found using the formula:
[tex]\[
P = \frac{75}{100} \times (n + 1)
\][/tex]
Here, [tex]\( n = 14 \)[/tex] (since there are 14 revenue figures).
So, the position [tex]\( P \)[/tex] is:
[tex]\[
P = \frac{75}{100} \times (14 + 1) = \frac{75}{100} \times 15 = 11.25
\][/tex]
3. Find the value at the 75th percentile position:
The 75th percentile value will be between the 11th and 12th values in the sorted data list.
Specifically, it is:
[tex]\[
0.25 \times (\text{12th value} - \text{11th value}) + \text{11th value}
\][/tex]
From the sorted list:
[tex]\[
11th \text{ value} = 43,520, \quad 12th \text{ value} = 45,270
\][/tex]
Therefore:
[tex]\[
\begin{aligned}
\text{75th percentile value} &= 0.25 \times (45,270 - 43,520) + 43,520 \\
&= 0.25 \times 1,750 + 43,520 \\
&= 437.5 + 43,520 \\
&= 43,957.5
\end{aligned}
\][/tex]
Hence, the amount that separates the lowest 75% of the data from the highest 25% of the data is [tex]\( \mathbf{43,097.5} \)[/tex].
