Question

The deciles of any distribution are the points that mark off the lowest 10% and the highest 10% of the distribution. the deciles of a density curve are therefore the points with an area 0.1 and 0.9 to their left under the curve.
a.what are the deciles of the standard normal distribution?
b.assuming that the distribution of men's heights is approximately normally distributed with a mean of 69.5 inches and a standard deviation of 2.7 inches, what are the deciles of this distribution?

Answer 1

(a) We need to figure out the deciles for a normal distribution curve in this part.

In order to do so, we need to find the z score corresponding to probability 0.10 and 0.90 on the normal distribution curve.

Upon looking at the normal distribution table, we can see that the probability P = 0.10 corresponds to a z score of -1.28 and probability P = 0.90 corresponds to a z score of 1.28.

Therefore, deciles are -1.28 and 1.28 for a normal distribution curve.

(b) In this part, we are given that mean height is 69.5 inches and standard deviation in height is 2.7 inches. Therefore, we can find the deciles using the z score formula as shown below:

$z=\frac{x-\mu}{\sigma }\Rightarrow \pm 1.28=\frac{x-69.5}{2.7}\\ x-69.5 = \pm 3.456\Rightarrow x=69.5\pm 3.456\Rightarrow \\ x=66.044,72.956$

Therefore, the required deciles are 66.044 and 72.956