Question

Given the scores on a certain exam are normally distributed with a mean of 75 and a standard deviation of 5

a. Calculate the z-score for 80. Find the percentage of students with scores above 80

b. Calculate the z-score for 60. Find the percentage of students with scores below 60.
c. Calculate the z-scores for 70 and 90. Find the percentage of students with scores between 70 and 90.
d. What is the median?
e. What test score value has a Z-score of -2.25?

f. What test score is the 85th percentile?

Answer 1

Let X denote the random variable representing the scores on a certain exam. Now, we are given that:

Moreover, the formula for z-score for a given score x is given by:

a.

The z-score corresponding to the score of 80 is given by:

The percentage of students with scores above 80 is given by:

b.

The z-score corresponding to the score of 60 is given by:

The percentage of students with scores below 60 is given by:

c.

The z-score corresponding to the score of 70 is given by:

The z-score corresponding to the score of 90 is given by:

The percentage of students with scores between 70 and 90 is given by:

d.

We know that a normal distribution is symmetric, thus its mean and median are equal. Thus, we get:

Median = Mean = μ = 75 [ANSWER]