Question

20. Sketch the normal curve having the parameters

a. $\mu =-1$and$\sigma =2$

b. $\mu =3$and$\sigma =2$

c. $\mu =-1$and$\sigma =0.5$

Short answer

a. The graph of the parameter $\mu =-1\text{and}\sigma =2$is:

b. The graph of the parameter $\mu =3\text{and}\sigma =2$is:

c. The graph of the parameter $\mu =-1\text{and}\sigma =0.5$is:

Step-by-step solution

Part (a) Step 1: Given Information

Sketch the normal curve having the parameters:

Mean $\mu =-1$

Standard deviation$\sigma =2$

Part (a) Step 2: Explanation

The probability distribution curve of a normal random variable is called a normal curve.

It's a representation of a normal distribution in graphical form.

If $X$is a continuous random variable with mean $\mu$and standard deviation $\sigma$, then a normal curve with random variable localid="1650897073505" $X$has the following equation:

localid="1650897079262" $f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{-(x-\mu {)}^2}{2{\sigma }^2}};-\mathrm{\infty }<x<\mathrm{\infty },-\mathrm{\infty }<\mu <\mathrm{\infty },\sigma >0.$

Furthermore, a normal curve with random variable localid="1650897086659" $Z$has the following equation:

localid="1650897092286" $f\left(z\right)=\frac{1}{\sqrt{2\pi }}{e}^{\frac{-z^2}{2}}$

localid="1650897100801" $Z$has a mean of localid="1650897108373" $0$and a standard deviation of localid="1650897115084" $1$, correspondingly.

The population mean localid="1650897121663" $\mu$and the population standard deviation localid="1650897128400" $\sigma$are two population metrics that are commonly included in a normal curve.

The bell-shaped normal distribution with localid="1650897134454" $\mu =-1$and localid="1650897140033" $\sigma =2$depicts the area under the normal distribution curve.

Arealocalid="1650897146320" $(probability)=0.3085$

Part (b) Step 3: Given Information

Sketch the normal curve having the parameters:

Mean$\mu =3$

Standard deviation$\sigma =2$

Part (b) Step 4: Explanation

The probability distribution curve of a normal random variable is called a normal curve. It's a representation of a normal distribution in graphical form.

If $X$is a continuous random variable with mean $\mu$and standard deviation $\sigma$, then a normal curve with random variable $X$has the following equation:

$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{-(x-\mu {)}^2}{2{\sigma }^2}};-\infty <x<\infty ,-\infty <\mu <\infty ,\sigma >0$

Furthermore, a normal curve with random variable $Z$has the following equation:

$f\left(z\right)=\frac{1}{\sqrt{2\pi }}{e}^{\frac{-z^2}{2}}$

$Z$has a mean of $0$and a standard deviation of $1$, correspondingly.

The population mean $\mu$and the population standard deviation $\sigma$are two population metrics that are commonly included in a normal curve.

The bell-shaped normal distribution with $\mu =3$and $\sigma =2$depicts the area under the normal distribution curve.

Area (probability)$=0.9332$

Part (c) Step 5: Given Information

Sketch the normal curve having the parameters:

Mean$\mu =-1$

Standard deviation$\sigma =0.5$

Part (c) Step 6: Explanation

The probability distribution curve of a normal random variable is called a normal curve. It's a representation of a normal distribution in graphical form.

If $X$is a continuous random variable with mean $\mu$and standard deviation $\sigma$, then a normal curve with random variable $X$has the following equation:

$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{-(x-\mu {)}^2}{2{\sigma }^2}};-\infty <x<\infty ,-\infty <\mu <\infty ,\sigma >0$

Furthermore, a normal curve with random variable $Z$has the following equation:

$f\left(z\right)=\frac{1}{\sqrt{2\pi }}{e}^{\frac{-z^2}{2}}$

$Z$has a mean of $0$and a standard deviation of $1$, correspondingly.

The population mean $\mu$and the population standard deviation $\sigma$are two population metrics that are commonly included in a normal curve.

The bell-shaped normal distribution with $\mu =-1$and $\sigma =0.5$depicts the area under the normal distribution curve.

Area (probability)$=0.0228$