Chapter 7, Problem 1RE

(a)

To determine

To identify: The value of $\mu$

Answer to Problem 1RE

Solution: The value of $\mu$ is 60.

Explanation of Solution

Given: The figure for the probability distribution is provided in the question.

Explanation: $\mu$ denotes the mean of the distribution. The provided figure is the curve of the normal distribution. In the provided figure, the distribution is symmetric and the mode of the distribution is at the point 60. Therefore, the point 60 can be considered as the mean of the distribution.

(b)

To determine

To identify: The value of $\sigma$

Answer to Problem 1RE

Solution: The value of $\sigma$ is 10.

Explanation of Solution

Given: The figure for the probability distribution is provided in the question.

Explanation: $\sigma$ denotes the standard deviation of the distribution. The value of the standard deviation can be obtained from the inflection point of the curve. The inflection points for the normal distribution are $\mu -\sigma$ and $\mu +\sigma$, where the curvature is changed. In the provided figure, the two inflection points are

$\begin{array}{l}\mu -\sigma =50\\ \mu +\sigma =70\end{array}$

As the value of the mean or $\mu$ is 60, so the value of $\left(\sigma \right)$ is obtained from the inflection points and can be calculated as

$\begin{array}{c}\mu -\sigma =50\\ \sigma =60-50\\ \sigma =10\end{array}$

Conclusion: The provided figure represents the normal curve whose mean $\left(\mu \right)$ is 60 and $\left(\sigma \right)$ is 10.

(c)

To determine

To interpret: The area under the normal curve when the area right to $x=75$ is provided.

Answer to Problem 1RE

Solution: The provided information can be interpreted as the proportion of the value of the random variable 75 or more is 0.0668. It can be also interpreted as the probability that value of the random variable is 75 or more than 75 is 0.0668.

Explanation of Solution

Given: The figure for the probability distribution is provided in the question. The area covered by the normal curve to the right of the value $x=75$ is 0.0668.

Explanation: The provided figure represents the normal curve whose mean $\left(\mu \right)$ is 60 and standard deviation $\left(\sigma \right)$ is 10. The area covered under the normal curve provides the proportion of the value of the random variable as well as the probability of the occurrence of the random variable whose value is greater than or less than that value or in between two values.

On the basis of the provided information, the area covered under the normal curve can be interpreted as the proportion of the value of the random variable is 75 or more is 0.0668 and the probability that the value of the random variable is 75 or more is 0.0668.

(d)

To determine

To interpret: The area under the normal curve when the area right to $x=50\text{ and }x=75$ is provided.

Answer to Problem 1RE

Solution: The provided information can be interpreted as the proportion of the value of the random variable that is between 50 and 75 is 0.7745. It can be also interpreted as the probability that value of the random variable lies between 50 and 75 is 0.7745.

Explanation of Solution

Given: The figure for the probability distribution is provided in the question. The area covered by the normal curve between the values $x=50\text{ and }x=75$ is 0.7745.

Explanation: The provided figure represents the normal curve whose mean $\left(\mu \right)$ is 60 and standard deviation $\left(\sigma \right)$ is 10. The area covered under the normal curve provides the proportion of the value of the random variable as well as the probability of the occurrence of the random variable whose value is greater than or less than that value or in between two values.

On the basis of the provided information, the area covered under the normal curve can be interpreted as the proportion of the value of the random variable that takes values between 50 and 75 is 0.7745 and the probability that the value of the random variable lies between the value 50 and 75 is 0.7745.