Question

1. the salaries (in thousands) at a small company

25, 28, 30, 30, 32, 35, 37, 38, 39, 42, 45, 49, 54, 56
a. what percentile is the person earning $39,000?
b. what salary marks the 75th percentile?

Explanation:

Step1: Calculate the percentile for \(39000\)

First, count the number of data points \(n = 14\).
Count the number of data points less than or equal to \(39\) (since the data is in thousands). There are \(8\) data points (\(25,28,30,30,32,35,37,38\)) less than \(39\).
The formula for percentile \(P=\frac{k}{n}\times100\), where \(k\) is the number of data points less than or equal to the value and \(n\) is the total number of data points.
So \(P=\frac{8}{14}\times 100=\frac{800}{14}\approx57.14\)

Step2: Calculate the \(75^{th}\) percentile

The formula for the position of the \(p^{th}\) percentile is \(L=\frac{p}{100}\times n\). For \(p = 75\) and \(n=14\), \(L=\frac{75}{100}\times14 = 10.5\)
When \(L\) is not an integer, we use the formula \(x_L=x_{j}+0.5\times(x_{j + 1}-x_{j})\), where \(j\) is the integer part of \(L\). Here \(j = 10\), \(x_{10}=42\), \(x_{11}=45\)
\(x_{10.5}=42+0.5\times(45 - 42)=42 + 1.5=43.5\) (in thousands)

Answer:

a. The person earning \(39000\) is at approximately the \(57^{th}\) percentile.
b. The \(75^{th}\) percentile is \(43500\) (since \(43.5\times1000 = 43500\))