Q. 6.93 - Elementary Statistics

Question

A variable is normally distributed with mean $6$ and standard deviation $2$.

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

Part (b) Obtain and interpret the $85th$ percentile.

Part (c) Find the value that $65\%$ 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 $0.95$ and two outside areas of $0.025$. Interpret your answer.

Step-by-Step Solution

Answer

Part (a): A quarter of all observations are less than $4.66$, half of all observations are less than $6$, and seventy per cent of all observations are less than $7.34$.

Part (b):$85\%$ of all observations are smaller than $8.08$.

Part (c): The value that $65\%$ of all possible values of the variable exceeds is $5.22.$

Part (d):$95\%$ of all observations are between $2.08$ and $9.92$.

Part (a) Step 1: Given information

The mean is $6$ and standard deviation is $2.$

Part (a) Step 2: Determine the quartiles of the variable.

Sketch a normal curve using$$\begin{gathered} \mu =6, \\ \sigma =2 \end{gathered}$$

The z-score of the random variable x is $z=\frac{x-\mu }{\sigma }$.

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.

On finding the quartiles,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ \Rightarrow z\sigma =x-\mu \\ \Rightarrow x=\mu +z\sigma \end{gathered}$$

First quartile,

$Q_1=6+(-0.67)\times 2$

     $=4.66$

Second quartile,

$Q_2=6+\left(0\right)\times 2$

      $=6$

Third quartile,

$Q_3=6+(0.67)\times 2$

     $=7.34$

On interpreting, a $25\%$ of all observations are less than $4.66$, $50\%$of all observations are less than $6$, and  of all observations are less than $7.34$.

Part (b) Step 1: Interpret the 85 th percentile.

The data is divided into a hundred equal pieces using the percentiles. The area below the $85th$ percentile has a proportion of $0.85$.

The $z-$score for the proportion $0.85$ is $1.04$, according to Table II in Appendix A.

Find the $85th$ percentile,

$x=\mu +z\sigma$

$\Rightarrow P_2=6+(1.04)\times 2$

          $=8.08.$

On interpreting, we can say, a quarter of all observations are less than $4.66$, half of all observations are less than $6$ and seventy per cent of all observations are less than $7.34$.

Part (c) Step 1: Find the possible value that 65 % of all possible values of the variable exceed.

The z-score corresponding to a proportion in Table II of Appendix A is $-0.39$ is given below,

$1-0.65=0.35$

$x=\mu +z\sigma$

$\Rightarrow x=6+(-0.39)\times 2$

        $=5.22$

On interpreting the value that $65\%$of all possible values of the variable exceeds is $5.22$.

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

From Table II in the Appendix A, the $z-$scores corresponding to the proportions of areas below $0.025$, and $0.95$ are $-1.96$, and $1.96$ respectively.

$x=\mu +z\sigma$

$\Rightarrow x_1=6+(-1.96)\times 2$

          $=2.08$

$x_2=6+(1.96)\times 2$

     $=9.92$

On interpreting, we get,$95\%$ of all observation are between $2.08$ and $9.92$.