Question

A variable is normally distributed with mean $10$ and standard deviation $3$.

a. Determine and interpret the quartiles of the variable.

b. Obtain and interpret the seventh decile.

c. Find the value that $35\%$ 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 $0.99$ and two outside areas of $0.005$. Interpret your answer.

Short answer

a). The variable's quartiles are $7.99,10$, and $12.01$, and $25\%$of the observations are less than $7.99,50\%$less than $10$, and $75\%$less than $12.01$.

b). The $7^{\text{th}}$decile is equal to $11.572$and $70\%$of the observations are less than $0.524$.

C). $35\%$of all observations are larger than $8.845$.

d). $99\%$ of the observations are between $-1.997$ and $21.997$.

Step-by-step solution

Part (a) Step 1: Given Information

Given data:

A variable is normally distributed with a mean $10$and a standard deviation $3$.

Part (a) Step 2: Explanation

To find the $z-$scores corresponding to the percent $0.25,0.50$and $0.75$which are:

$-0.67,0,0.67$

Calculate the following value:

$x=\mu +z\sigma$

$=10+(-0.67)·3$

$=7.99$

$x=\mu +z\sigma$

$=10+\left(0\right)·3$

$=10$

$x=\mu +z\sigma$

$=10+(0.67)·3$

$=12.01$

Interpretation: $25\%$of the observations are less less than $7.99,50\%$are less than $10$, and $75\%$are less than$12.01$.

Part (b) Step 1: Given Information

Given data:

Mean $=10$.

Standard deviation$=3$.

Part (b) Step 2: Explanation

Since $D_7=P_{70}$, the $7$thdecile is equivalent to the $70$th percentile,

To find the $z$- scores corresponding to the percent ${70}^{\text{th}}$percentile :

$0.524$

Calculate the following value :

$x=\mu +z\sigma$

$=10+(0.524)·3$

$=11.572$

Interpretation: $70\%$of the observations are less than $0.524$.

Part (c) Step 1: Given Information

Given data:

Mean $=10$.

Standard deviation$=3$.

Part (c) Step 2: Explanation

To find the $z-$ scores corresponding to the percent $1-0.35=0.65$ which is:

$-0.385$

To calculate the corresponding value:

$x=\mu +z\sigma$

$=10+(-0.385)3$

$=8.845$

Interpretation: $35\%$of all observations are larger than $8.845$

Part (d) Step 1: Given Information

Given data:

Mean$=10$.

Standard deviation$=3$.

Part (d) Step 2: Explanation

To find the $z-$scores corresponding to the percent $0.005$which is:

$\pm 3.999$

To calculate the corresponding value:

$x=\mu +z\sigma$

$=10+(-3.999)·3$

$=-1.997$

$x=\mu +z\sigma$

$=10+(3.999)·3$

$=21.997$

Interpretation: $99\%$ of the observations are between $-1.997$ and $21.997$