Question

A variable is normally distributed with a mean $0$and a standard deviation $4$.

a. Determine and interpret the quartiles of the variable.

b. Obtain and interpret the second decile.

c. Find the value that $15\%$ of all possible values of the variable exceed.

d. Find the two values that divide the area under the corresponding normal curve into a middle area of \(0.80\) and two outside areas of $0.10$. Interoret vour answer.

Short answer

a). Quartiles: $Q_1=-2.68$,

$Q_2=0$,

$Q_3=2.68$.

b). The second decile is $-3.36$.

c). $15\%$of observation is $4.16$.

d). $80\%$of all observation are between$-5.12$and$5.12$.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

Mean $=0$.

Standard deviation$=4$.

Part (a) Step 2: Explanation

The data is divided into four equal sections by the quartiles. The areas below the first, second, and third quartiles have proportions of $0.25,0.5$, and $0.75$, respectively. The $z-$scores for the proportions $0.25,0.5$, and $0.75$are $-0.67,0$, and $0.67$, respectively, according to Table II in Appendix A.

Calculate the quartiles:

$z=\frac{x-\mu }{\sigma }$

The first quartile:

$Q_1=0+(-0.67)·4$

$=-2.68$

The second quartile:

$Q_2=0+\left(0\right)·4$

$=0$

The third quartile:

$Q_3=0+(0.67).4$

$=2.68$

Interpretation: $25\%$ of all observations are less than $-2.68,50\%$ of all observations are less than $0$, and $75\%$ of all observations are less than $2.68$.

Part (b) Step 1: Given Information

Given data:

Mean: $0$,

Standard deviation: $4$.

Part (b) Step 2: Explanation

The deciles divide the data into ten equal parts. The proportion of the area below the $2^{\text{nd}}$decile is $0.20$. From Table II in the Appendix A, the $z$- score corresponding to the proportion $0.20$is $-0.84$.

Calculate the $2^{\text{nd}}$decile:

$x=\mu +z\sigma$

$\Rightarrow D_2=0+(-0.84)·4$

localid="1653228069385" $=-3.36$

Interpretation:

Approximately $20\%$of all observations are less than $-3.36$.

Part (c) Step 1: Given Information

Given data:

Mean:$0$

Standard deviation:$4$.

Part (c) Step 2: Explanation

The $z-$score corresponding to a proportion of $1-0.15=0.85$in Table II of Appendix A is $1.04$.

Calculate corresponding value:

$x=\mu +z\sigma$

$\Rightarrow x=0+(1.04)·4$

$=4.16$

Part (d) Step 1: Given Information

Given data:

Mean:$0$.

Standard deviation:$4$.

Part (d) Step 2: Explanation

The $z$- scores for the proportions of regions below $0.10$and $0.80$, respectively, are $-1.28$and$1.28$, according to Table II in Appendix A.

Determine corresponding values:

$x=\mu +z\sigma$

$\Rightarrow x_1=0+(-1.28)·4$

$=-5.12$

$x_2=0+(1.28)·4$

$=5.12$

Interpretation:

$80\%$ of all observation are between $-5.12$ and $5.12$.