Chapter 17, Problem 1E

Consider the normal distribution represented by the normal curve in Fig 17-12. Assume that P is a point of inflection of the curve.

Figure 17-12

a. Find the mean $\mu$ of the distribution.

b. Find the median M of the distribution.

c. Find the standard deviation $\sigma$ of the distribution.

To determine

(a)

To find:

The mean $\mu$ of the distribution.

Answer to Problem 1E

Solution:

The mean $\mu$ of the distribution is $80\text{lb}$.

Explanation of Solution

Given:

The normal distribution curve is,

With P as a point of inflection of the curve.

Definition:

The point of intersection of the horizontal (data) axis and the line of symmetry of the normal curve, the center of the distribution. The center corresponds to both the median and mean of the distribution.

Calculation:

From the given figure, it is seen that the point of intersection of the horizontal axis and the line of symmetry is at $80\text{lb}$.

This implies that the mean $\mu$ of the distribution is $\mu =80\text{lb}$.

Conclusion:

The mean $\mu$ of the distribution is $80\text{lb}$.

To determine

(b)

To find:

The median M of the distribution.

Answer to Problem 1E

Solution:

The median M of the distribution is $80\text{lb}$.

Explanation of Solution

Given:

The normal distribution curve is,

With P as a point of inflection of the curve.

Definition:

The point of intersection of the horizontal (data) axis and the line of symmetry of the normal curve, the center of the distribution. The center corresponds to both the median and mean of the distribution.

Calculation:

From the given figure, it is seen that the point of intersection of the horizontal axis and the line of symmetry is at $80\text{lb}$.

This implies that the median M of the distribution.is $80\text{lb}$.

Conclusion:

The median M of the distribution is $80\text{lb}$.

To determine

(c)

To find:

The standard deviation $\sigma$ of the distribution.

Answer to Problem 1E

Solution:

The standard deviation $\sigma$ of the distribution is $10\text{lb}$.

Explanation of Solution

Given:

The normal distribution curve is,

With P as a point of inflection of the curve.

Definition:

The standard deviation of the normal distribution is the horizontal distance between the axis of symmetry of the curve and one of the two points of inflection.

Calculation:

From the given figure, it is seen that the point of intersection of the horizontal axis and the line of symmetry is at $80\text{lb}$.

The point of inflection of the curve is $90\text{lb}$.

Therefore the distance between the point of inflection and the line of symmetry is,

$90-80=10\text{lb}$.

This implies that the standard deviation $\sigma$ of the distribution.is $10\text{lb}$.

Conclusion:

The standard deviation $\sigma$ of the distribution.is $10\text{lb}$.