Question

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

Answer 1

Answer:

The 7th percentile is 34.2

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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.

$Z = \frac{X - \mu}{\sigma}$

$-1.475 = \frac{X - 46}{8}$

$X - 46 = -1.475*8$

$X = 34.2$

The 7th percentile is 34.2