Question

1. the average weekly earnings in dollars for various industries are listed below.

479 506 680 620 710 490 800 550

a) find the percentile rank of the person that earns $550
b) find the percentile rank of the person that earns $710
c) find the data value that corresponds to the 40th percentile
d) find the data value that corresponds to the 80th percentile

Explanation:

Step1: Sort the data

First, sort the data set \(479, 506, 680, 620, 710, 490, 800, 550\) in ascending order.
\(479,490,506,550,620,680,710,800\)

Step2: Calculate percentile rank formula

The formula for percentile rank \(PR=\frac{CF + 0.5\times F}{n}\times100\), where \(CF\) is the number of data values less than the given value, \(F\) is the number of data values equal to the given value, and \(n\) is the total number of data values. Here \(n = 8\)

Part a:

For \(x = 550\), \(CF=3\), \(F = 1\)
\(PR=\frac{3+0.5\times1}{8}\times100=\frac{3.5}{8}\times100 = 43.75\)

Part b:

For \(x = 710\), \(CF = 6\), \(F=1\)
\(PR=\frac{6 + 0.5\times1}{8}\times100=\frac{6.5}{8}\times100=81.25\)

Step3: Calculate data value from percentile formula

The formula for finding the data value corresponding to a percentile \(L=\frac{P}{100}\times n\), where \(P\) is the percentile and \(n\) is the number of data values. Then, if \(L\) is an integer, the data value is the average of the \(L\) - th and \((L + 1)\) - th ordered data values. If \(L\) is not an integer, round up to the next integer.

Part c:

For \(P = 40\), \(L=\frac{40}{100}\times8=3.2\). Round up \(L\) to \(4\). The \(4\) - th ordered data value is \(550\)

Part d:

For \(P=80\), \(L=\frac{80}{100}\times8 = 6.4\). Round up \(L\) to \(7\). The \(7\) - th ordered data value is \(710\)

Answer:

a) \(43.75\)
b) \(81.25\)
c) \(550\)
d) \(710\)