Chapter 6, Problem 1CQ

Following is a probability density curse for a population.

1. a. What proportion of the population is between 2 and 4?
2. b. If a value is chosen at random from this population, what is the probability that it will be greater than 2?

To determine

Find the proportion of the population is between 2 and 4.

Answer to Problem 1CQ

The proportion of the population is between 2 and 4 is 0.32.

Explanation of Solution

Calculation:

The given probability density curve indicates the area between 0 and 2, and the area between 4 and 10. The area between 0 and 2 is 0.59 and the area between 4 and 10 is 0.09.

The proportion of the population is between 2 and 4 represents the area under the curve between 2 and 4.

Generally total area of the curve is 1.

The proportion of the population is between 2 and 4 is obtained as follows.

$\begin{array}{c}P\left(\text{Between 2 and 4}\right)=\text{Total area}-P\left(\text{Between 0 and 2}\right)-P\left(\text{Between 4 and 10}\right)\\ =1-0.59-0.09\\ =1-0.68\\ =0.32\end{array}$

Therefore, the proportion of the population is between 2 and 4 is 0.32.

To determine

Find the probability that a randomly selected value will be greater than 2.

Answer to Problem 1CQ

The probability that a randomly selected value will be greater than 2 is 0.41.

Explanation of Solution

Calculation:

From part (a), the area between 2 and 4 is 0.32. Also, the area between 0 and 2 is 0.59 and the area between 4 and 10 is 0.09.

The probability that a randomly selected value will be greater than 2 represents the area to the right of 2.

The probability that a randomly selected value will be greater than 2 is obtained as follows.

$\begin{array}{c}P\left(\text{Greater than 2}\right)=P\left(\text{Between 2 and 4}\right)+P\left(\text{Between 4 and 10}\right)\\ =0.32+0.09\\ =0.41\end{array}$

Thus, the probability that a randomly selected value will be greater than 2 is 0.41.