91Ó°ÊÓ

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

Sketch a normal curve using$$\begin{gathered} \mathrm{μ}=6, \\ \mathrm{σ}=2 \end{gathered}$$

The z-score of the random variable x is $z=\frac{x-\mathrm{μ}}{\mathrm{σ}}$.

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,

$$\begin{gathered} z=\frac{x-\mathrm{μ}}{\mathrm{σ}} \\ ⇒z\mathrm{σ}=x-\mathrm{μ} \\ ⇒x=\mathrm{μ}+z\mathrm{σ} \end{gathered}$$

First quartile,

$Q_1=6+(-0.67)×2$

$=4.66$

Second quartile,

$Q_2=6+\left(0\right)×2$

$=6$

Third quartile,

$Q_3=6+(0.67)×2$

$=7.34$

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 $7.34$.

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,

$x=\mathrm{μ}+z\mathrm{σ}$

$⇒P_2=6+(1.04)×2$

$=8.08.$

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$7.34$.

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

$1-0.65=0.35$

$x=\mathrm{μ}+z\mathrm{σ}$

$⇒x=6+(-0.39)×2$

$=5.22$

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

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.

$x=\mathrm{μ}+z\mathrm{σ}$

$⇒x_1=6+(-1.96)×2$

$=2.08$

$x_2=6+(1.96)×2$

$=9.92$

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