Question

Deciles The deciles of any distribution are the points that mark off the lowest $10\%$and the highest $10\%$. The deciles of a density curve are therefore the points with area $0.1$and $0.9$to their left under the curve.

(a) What are the deciles of the standard Normal distribution?

(b) The heights of young women are approximately Normal with mean $64.5$inches and standard deviation $2.5$inches. What are the deciles of this distribution? Show your work

Short answer

a) The declines of the standard normal distribution is $\pm 1.28$

b)The declines of the young women are$61.3$inches and$67.7$inches

Step-by-step solution

Part (a) Step 1: Given Information

The deciles of any lowest distribution$=10\%$

The deciles of any highest distribution$=10\%$

Areas$=0.1$and$0.9$

Part (a) Step 2: Explanation

The figure illustrates what we're looking for:

In the standard Normal Curve, the deciles are $10\%$and $90\%$percent of the distribution on either side of the standard curve.

Due to the curvature's symmetry, the area from $z=0$to either end is $0.5$while the area from $z=0$to the top or bottom is $1$.

As shown here, $0.1$is equal to $0.5-0.4$and $0.9$is equal to $0.5+0.4$.

Based on the Table of Normal Curve, $0.3997$is the nearest value to $0.4$.

This corresponds to a $z=1.28$value.

Therefore, $z=-1.28$is the standardized value with area $0.10$to the left of $z=1.28$and is the standardized value with area $0.90$to the left of $z=1.28$.

Hence, the deciles are$\pm 1.28$.

Part (b) Step-1: Given Information

The heights of young women with mean$=64.5inches$

Standard deviation $=2.5inches$

we have to find out the the declines of the heights of young women.

Part (b) Step-2: Explanation

Assuming random variable $X$indicates heights of young women, then we are given the following:

$x~N(\mu ,{\sigma }^2)where\mu =64.5inchesand\sigma =2.5inches$

The unknown $x\text{'}s$standardized value is $z=\pm 1.28$.

Hence, $X$satisfies the equation:

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ \pm 1.28=\frac{x-64.5}{2.5} \end{gathered}$$

Find the$x$value:

$$\begin{gathered} x=64.5\pm 1.28(2.5) \\ x=61.3inchesandx=67.7inches \end{gathered}$$

The decline for the heights of young women are $61.3inchesand67.7inches$