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

Answer:

Q3 = 65.7825.

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:

$\mu = 63.6, \sigma = 2.5$

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.

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

$0.675 = \frac{X - 63.6}{2.5}$

$X - 63.6 = 0.675*2.5$

$X = 65.7825$

Q3 = 65.7825.