Q. 20 - Elementary Statistics

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 $\sigma =0.5$

Step-by-Step Solution

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:

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 $X$ has the following equation:

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

Furthermore, a normal curve with random variable $Z$ 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 $\sigma =2$ depicts the area under the normal distribution curve.

Area$( probability)=0.3085$

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)$=0.0228$