Question

1. for the table of frequency below:

x f
45 - 47 2
42 - 44 1
39 - 41 3
36 - 38 14
33 - 35 15
30 - 32 12
27 - 29 9
24 - 26 7
21 - 23 4
a. what is the percentile rank of a person with a score of 31?
b. what is the percentile rank of a person with a score of 42?
c. what is the percentile rank of a person with a score of 26?
d. what is the percentile rank of a person with a score of 41?
e. what is the percentile rank of a person with a score of 23? (note that cf below is \0\)

Explanation:

Step1: Calculate total frequency

First, find the sum of all frequencies. $N=2 + 1+3 + 14+15+12+9+7+4=77$

Step2: Determine the class - interval containing the score

For a score of 31, it lies in the class - interval 30 - 32. The lower limit of this class $L = 30$, the frequency of this class $f=12$, the cumulative frequency of the classes below $30 - 32$ $cf_b=4 + 7+9=20$, and the class width $w = 3$.

Step3: Use the percentile - rank formula

The formula for percentile rank $PR=\frac{cf_b+\frac{(X - L)}{w}\times f}{N}\times100$. Substituting $X = 31$, $L = 30$, $f = 12$, $cf_b=20$, $w = 3$ and $N = 77$:
$PR=\frac{20+\frac{(31 - 30)}{3}\times12}{77}\times100=\frac{20 + 4}{77}\times100=\frac{24}{77}\times100\approx31.17$

For a score of 42, it lies in the class - interval 42 - 44. $L = 42$, $f = 1$, $cf_b=2+3+14+15+12+9+7+4=66$, $w = 3$.
$PR=\frac{66+\frac{(42 - 42)}{3}\times1}{77}\times100=\frac{66}{77}\times100\approx85.71$

For a score of 26, it lies in the class - interval 24 - 26. $L = 24$, $f = 7$, $cf_b=4$, $w = 3$.
$PR=\frac{4+\frac{(26 - 24)}{3}\times7}{77}\times100=\frac{4+\frac{14}{3}}{77}\times100=\frac{\frac{12 + 14}{3}}{77}\times100=\frac{\frac{26}{3}}{77}\times100=\frac{26}{231}\times100\approx11.26$

For a score of 41, it lies in the class - interval 39 - 41. $L = 39$, $f = 3$, $cf_b=2+1+14+15+12+9+7+4=64$, $w = 3$.
$PR=\frac{64+\frac{(41 - 39)}{3}\times3}{77}\times100=\frac{64 + 2}{77}\times100=\frac{66}{77}\times100\approx85.71$

For a score of 23, it lies in the class - interval 21 - 23. $L = 21$, $f = 4$, $cf_b=0$, $w = 3$.
$PR=\frac{0+\frac{(23 - 21)}{3}\times4}{77}\times100=\frac{\frac{8}{3}}{77}\times100=\frac{8}{231}\times100\approx3.46$

Answer:

a. Approximately 31.17
b. Approximately 85.71
c. Approximately 11.26
d. Approximately 85.71
e. Approximately 3.46