Question

Consider a normal distribution that has N(55,7) (that is, a mean of 55 and a standard deviation of 7). To the nearest integer, the percentile rank of a value of 50 in this distribution is: A. 76 B. 48 C. 43 D. 24 E. 71

Answer 1

D. 24

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 55, \sigma = 7`

To the nearest integer, the percentile rank of a value of 50 in this distribution is:

This is the pvalue of Z when X = 50. So

`Z = (X - \mu)/(\sigma)`

`Z = (50 - 55)/(7)`

`Z = -0.71`

`Z = -0.71` has a pvalue of 0.2388.

So the correct answer is:

D. 24