Question

Assume the random variable X is normally​ distributed, with mean 46 and standard deviation 8. Find the 7 th percentile.

Answer 1

The 7th percentile is 34.2

Explanation:

Problems of normally distributed samples are 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 = 46, \sigma = 8`

Find the 7 th percentile.

The value of X when Z has a pvalue of 0.07. So it is X when Z = -1.475.

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

`-1.475 = (X - 46)/(8)`

`X - 46 = -1.475*8`

`X = 34.2`

The 7th percentile is 34.2