Q. 12 - Elementary Statistics

Question

12. Consider the normal curves that have the parameters $\mu =1.5$ and $\sigma =3;\mu =1.5$ and $\sigma =6.2;\mu =-2.7$ and $\sigma =3;\mu =0$ and $\sigma =1.$

a. Which curve has the largest spread?

b. Which curves are centered at the same place?

c. Which curves have the same spread?

d. Which curve is centered farthest to the left?

e. Which curve is the standard normal curve?$\sigma$

Step-by-Step Solution

Answer

a. The curve which has the largest spread parameter is $\mu =1.5$ and $\sigma =6.2$

b. The curves which are centered at the same place is $\mu =1.5$and $\sigma =3$ and $\mu =1.5$ and $\sigma =6.2.$

c. The curves which have the same spread is $\mu =1.5$ and $\sigma =3$ and $\mu =-2.7$and $\sigma =3.$

d. The curve which is centered farthest to the left is $\mu =-2.7$and $\sigma =3$

e. The curve which is the standard normal curve is $\mu =0$ and $\sigma =1$

Part (a) Step 1: Given Information

Consider the normal curve with parameters:

$\begin{array}{rl}-\mu & =1.5 \text{and }\sigma =3\\ -\mu & =1.5 \text{and }\sigma =6.2\\ -\mu & =-2.7 \text{and }\sigma =3\\ -\mu & =0 \text{and }\sigma =1\end{array}$

Part (a) Step 2: Explanation

The standard deviation $\sigma$of the curve with the biggest spread is the largest. The curve having the largest spread among the given curves is:

$\mu =1.5$ and $\sigma =6.2$

Part (b) Step 3: Given Information

Consider the normal curve with parameters:

$\begin{array}{rl}-\mu & =1.5 \text{and }\sigma =3\\ -\mu & =1.5 \text{and }\sigma =6.2\\ -\mu & =-2.7 \text{and }\sigma =3\\ -\mu & =0 \text{and }\sigma =1\end{array}$

Part (b) Step 4: Explanation

The mean $\mu$ of curves with the same centers is the same. The curves that are centered in the same position among the given curves are:

$\mu =1.5$ and $\sigma =3$ and $\mu =1.5$and $\sigma =6.2$

Part (c) Step 5: Given Information

Consider the normal curve with parameters:

$\begin{array}{rl}-\mu & =1.5 \text{and }\sigma =3\\ -\mu & =1.5 \text{and }\sigma =6.2\\ -\mu & =-2.7 \text{and }\sigma =3\\ -\mu & =0 \text{and }\sigma =1\end{array}$

Part (c) Step 6: Explanation

The standard deviation $\sigma$ of curves with similar spreads is the same. The curves with the same spreads among the given curves are:

$\mu =1.5\text{ and }\sigma =3\text{ and }\mu =-2.7\text{ and }\sigma =3$

Part (d) Step 7: Given Information

Consider the normal curve with parameters

$\begin{array}{rl}-\mu & =1.5 \text{and }\sigma =3\\ -\mu & =1.5 \text{and }\sigma =6.2\\ -\mu & =-2.7 \text{and }\sigma =3\\ -\mu & =0 \text{and }\sigma =1\end{array}$

Part (d) Step 8: Explanation

When the mean is smallest, the curve will be centered to the left. From the given curves, the curve that is centered to the left is:

$\mu =-2.7\text{ and }\sigma =3$

Part (e) Step 9: Given Information

Consider the normal curve with parameters:

$\begin{array}{rl}-\mu & =1.5 \text{and }\sigma =3\\ -\mu & =1.5 \text{and }\sigma =6.2\\ -\mu & =-2.7 \text{and }\sigma =3\\ -\mu & =0 \text{and }\sigma =1\end{array}$

Part (e) Step 10: Explanation

For each curve, the mean is $\mu =0$  and the standard deviation is $\sigma =1$. For each curve, the standard normal curve is:

$\mu =0\text{ and }\sigma =1$