The following frequency distribution presents a set of exam scores for a class of N= 20 students.
X F CF % C%
90-99 4 20 20 100
80-89 7 16 35 80
70-79 4 9 20 45
60-69 3 5 15 25
50-89 2 2 10 10
a)Using interpolation find the 30th percentile.
The answer I came up with was 72.
B) FIND THE 88TH PERCENTILE
93.5
C) WHAT IS THE PERCENTILE RANK FOR X=77
40th percentile
D) WHAT IS THE PERCENTILE RANK FOR X=90
81st percentile
-Are my answers correct. I did the % portion of the table myself. I'm not sure if I did it correctly though.
Thank You
I realize the table looks off. X= the scores 93-95...etc. cf= 4,7,4,3,2,. %=20,35,20,15,10. c%=100,80,45,25,10.
To find the answers, we need to understand the steps for interpolation and calculate the percentile rank for each given score in the frequency distribution.
a) To find the 30th percentile, we need to identify the interval that contains the 30th percentile rank. This can be done by calculating the cumulative frequency (CF) for each interval and locating the interval whose CF is greater than or equal to 30%. In this case, the interval is 70-79.
To find the 30th percentile within this interval, we use the percentile formula:
Percentile = L + ((P - CF) / f) * i
Where:
L = lower limit of the interval (70)
P = target percentile rank (30%)
CF = cumulative frequency of the previous interval (9)
f = frequency of the current interval (4)
i = width of the interval (10)
Substituting the values into the formula:
Percentile = 70 + ((30 - 9) / 4) * 10
= 70 + (21 / 4) * 10
= 70 + 52.5
= 122.5
So, the 30th percentile is approximately 122.5.
Based on this calculation, your answer of 72 is incorrect.
b) To find the 88th percentile, we follow the same steps. Looking at the cumulative frequency column, the interval which contains the 88th percentile is 90-99.
Using the percentile formula:
Percentile = L + ((P - CF) / f) * i
Where:
L = lower limit of the interval (90)
P = target percentile rank (88%)
CF = cumulative frequency of the previous interval (20)
f = frequency of the current interval (4)
i = width of the interval (10)
Substituting the values into the formula:
Percentile = 90 + ((88 - 20) / 4) * 10
= 90 + (68 / 4) * 10
= 90 + 170
= 260
So, the 88th percentile is 260.
Your answer of 93.5 is incorrect for the 88th percentile.
c) To find the percentile rank for X = 77, we need to identify the interval that contains the score 77. Based on the given frequency distribution, the interval is 70-79.
The percentile rank for X = 77 can be calculated using the formula:
Percentile Rank = (CF + ((f / i) * (X - L))) / N * 100
Where:
CF = cumulative frequency of the previous interval (9)
f = frequency of the current interval (4)
i = width of the interval (10)
X = score (77)
L = lower limit of the interval (70)
N = total number of scores (20)
Substituting the values into the formula:
Percentile Rank = (9 + ((4 / 10) * (77 - 70))) / 20 * 100
= (9 + (0.4 * 7)) / 20 * 100
= (9 + 2.8) / 20 * 100
= 11.8 / 20 * 100
= 0.59 * 100
= 59
So, the percentile rank for X = 77 is 59.
Your answer of 40th percentile is incorrect for X = 77.
d) To find the percentile rank for X = 90, we need to identify the interval that contains the score 90. Based on the given frequency distribution, the interval is 90-99.
The percentile rank for X = 90 can be calculated using the same formula as in part (c):
Percentile Rank = (CF + ((f / i) * (X - L))) / N * 100
Where:
CF = cumulative frequency of the previous interval (20)
f = frequency of the current interval (4)
i = width of the interval (10)
X = score (90)
L = lower limit of the interval (90)
N = total number of scores (20)
Substituting the values into the formula:
Percentile Rank = (20 + ((4 / 10) * (90 - 90))) / 20 * 100
= (20 + (4 / 10) * 0) / 20 * 100
= (20 + 0) / 20 * 100
= 20 / 20 * 100
= 1 * 100
= 100
So, the percentile rank for X = 90 is 100.
Your answer of 81st percentile is incorrect for X = 90.
In summary, only the answer for part (d) is incorrect. The correct answers are:
a) The 30th percentile is approximately 122.5.
b) The 88th percentile is 260.
c) The percentile rank for X = 77 is 59.
d) The percentile rank for X = 90 is 100.