Question

Assume that women have heights that are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. Find the value of the quartile Q3. (Hint: Q3 has an area of 0.75 to its left.)

Answer 1

Q3 = 65.7825.

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 = 63.6, \sigma = 2.5`

Find the value of the quartile Q3. (Hint: Q3 has an area of 0.75 to its left.)

This is the value of X when Z has a pvalue of 0.75. So it is X when Z = 0.675.

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

`0.675 = (X - 63.6)/(2.5)`

`X - 63.6 = 0.675*2.5`

`X = 65.7825`

Q3 = 65.7825.