Q. 6.96 - Elementary Statistics

Question

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

Part (a): Determine and interpret the quartiles of the variable.

Part (b): Obtain and interpret the second decile.

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

Part (d): Find the two values that divide the area under the corresponding normal curve into a middle area of $0.8$ and two outside areas of $0.1$. Interpret your answer.

Step-by-Step Solution

Answer

Part (a): The first percentile is $-2.698$. $25\%$of the x-values are below $-2.698$ and $75\%$ of the x-values are above $-2.698$.

The second percentile is $0$. $50\%$ of the x-values are below $0$ and $50\%$of the x-values are above $0$.

The third percentile is $2.698$. $75\%$ of the x-values are below $2.698$ and $25\%$of the x-values are above $2.698$.

Part (b): $20\%$of all observations are smaller than $-3.3665$.

Part (c): $4$ values of X exceeds $15\%$.

Part (d): $80\%$of all observations are between $-5.128$ and $5.128$.

Part (a) Step 1. Given information.

The given mean is $0$ and standard deviation is $4$.

Part (a) Step 2. Determine the first and second quartiles of the variable.

The first quartile or $25$th percentile of X is defined as $P\left(X\le x\right)=0.25$.

$$\begin{gathered} P\left(\frac{X-\mu }{\sigma }\le \frac{x-\mu }{\sigma }\right)=0.25 \\ P\left(Z\le \frac{x-0}{4}\right)=0.25 \\ \frac{x-0}{4}=-0.6745 \\ x=-2.698 \end{gathered}$$

On interpreting, we can say, $25\%$of the x-values are below $-2.698$ and $75\%$of the x-values are above $-2.698$.

The second quartile or $50$th percentile of X is defined as $P\left(X\le x\right)=0.5$.

$$\begin{gathered} P\left(\frac{X-\mu }{\sigma }\le \frac{x-\mu }{\sigma }\right)=0.5 \\ P\left(Z\le \frac{x-0}{4}\right)=0.5 \\ \frac{x-0}{4}=0 \\ x=0 \end{gathered}$$

On interpreting, we can say, $50\%$ of the x-values are below $0$ and $50\%$ of the x-values are above $0$.

Part (a) Step 3. Determine the third quartiles of the variable.

The third quartile or $75$th percentile of X is defined as $P\left(X\le x\right)=0.75$.

$$\begin{gathered} P\left(\frac{X-\mu }{\sigma }\le \frac{x-\mu }{\sigma }\right)=0.75 \\ P\left(Z\le \frac{x-0}{4}\right)=0.75 \\ \frac{x-0}{4}=0.6745 \\ x=2.698 \end{gathered}$$

On interpreting, we can say, $75\%$of the x-values are below $2.698$ and $25\%$of the x-values are above $2.698$.

Part (b) Step 1. Determine the second decile.

Shade the region corresponding second decile or $20$th percentile of X is defined as $P\left(X\le x\right)=0.2$.

$$\begin{gathered} P\left(\frac{X-\mu }{\sigma }\le \frac{x-\mu }{\sigma }\right)=0.2 \\ P\left(Z\le \frac{x-0}{4}\right)=0.2 \\ \frac{x-0}{4}=-0.8416 \\ x=-3.3665 \end{gathered}$$

On interpreting, we can say, $20\%$ of all observations are smaller than $-3.3665$.

Part (c) Step 1. Determine the value that 15 % of all possible values of the variable exceed.

On calculating the value,

$$\begin{gathered} P\left(X\ge x\right)=0.15 \\ 1-P\left(\frac{X-\mu }{\sigma }\le \frac{x-\mu }{\sigma }\right)=0.15 \\ P\left(Z\le \frac{x-0}{4}\right)=1-0.15 \\ \frac{x-0}{4}=1.036 \\ x=4.1457 \end{gathered}$$

On interpreting, approximately $4$ values of X exceeds $15\%$.

Part (d) Step 1. Find the two values.

The Z-score corresponding to the two outside areas of $0.1$ is given below,

$$\begin{gathered} z_1=-1.282, \\ {z}_2=1.282 \end{gathered}$$

Now, first value $x_1$ is obtained as,

$$\begin{gathered} x_1=0+\left(-1.282\right)4 \\ =-5.128 \end{gathered}$$

Second value $x_2$ is obtained as,

$$\begin{gathered} x_2=0+\left(1.282\right)4 \\ =5.128 \end{gathered}$$

On interpreting, we can say, $80\%$of all observations are between $-5.128$ and $5.128$.