Question

Suppose that the distribution of scores on an exam is mound shaped and approximately symmetric. The exam scores have a mean of 100 and the 16th percentile is 75. (Use the Empirical Rule.)

(a) What is the 84th percentile?

(b) What is the approximate value of the standard deviation of exam scores?

(c) What is the z-score for an exam score of 90?

(d) What percentile corresponds to an exam score of 150?

%

(e) Do you think there were many scores below 25? Explain. Since a score of 25 is three standard deviations below the mean, that corresponds to a percentile of   %. Therefore, there were few scores below 25.

Answer 1

Solution:
Given in the question
Mean = 100
Solution(a)
16th percentile is 75 which is below 25 than 100. So as per normal distribution mean of the data which distribute data in 50% to the left and 50% to the right of mean. so 84th percentile is (100+25) = 125
So 84th percentile is 125
Solution(b)
p-value = 0.16, so Z-score from Z table is -1
Z-score = (X-mean)/SD
-1 = (75-100)/SD
SD = 25
Solution(c)
Z-score form X=90 can be calculated as
Z-score = (90-75)/25 = 0.6
Solution(d)
If X = 150, than
Z = (150-100)/25 = 2
From Z table is p-value = 0.9772
So 97.72 percentile corresponds to an exam score of 150.
Solution(e)
If X=25, than Z= (25-100)/25 = -3
From Z score we found p-value is 0.00135
So I think that there is not many score below 25.