Chapter 6.3, Problem 98E

a.

To determine

To find: The probability that lie detector indicates that at least 10 of the people are honest.

Answer to Problem 98E

The probability is 0.5583

Explanation of Solution

Given:

Probability of success ( p ) = 0.2

Number of trials ( n ) = 12

Formula used:

The formula to compute the mean and standard deviation are:

  $\begin{array}{l}\text{Mean}\left(\mu \right)=n\times p\\ \text{Standard deviation}\left(\sigma \right)=\sqrt{n\times p\times \left(1-p\right)}\end{array}$

Calculation:

Consider, Y be the random that represents the number of people who are indicated as deceptive by lie detector

The probability can be calculated as:

  $\begin{array}{c}P\left(Y\le 3\right)=P\left(Y=0\right)+P\left(Y=1\right)+P\left(Y=2\right)\\ ={\displaystyle \sum _{r=0}^2{}^{12}{C}_r\times {\left(0.2\right)}^r\times {\left(1-0.2\right)}^{12-r}}\\ =0.5583\end{array}$

Thus, the required probability is 0.5583.

b.

To determine

To find: The mean of Y and interpret it.

Answer to Problem 98E

The mean is 2.4.

Explanation of Solution

The mean of Y can be calculated as:

  $\begin{array}{c}{\mu }_Y=n\times p\\ =12\left(0.2\right)\\ =2.4\end{array}$

Thus, the mean is 2.4.

Interpretation:

The computed value of mean shows that on an average out of 12 people , 2.4 people will be detected as deceptive by the lie detector.

c.

To determine

To find: The standard deviation of Y and interpret it.

Answer to Problem 98E

The standard deviation is 1.386.

Explanation of Solution

The standard deviation of Y can be calculated as:

  $\begin{array}{c}{\sigma }_Y=\sqrt{n\times p\times \left(1-p\right)}\\ =\sqrt{12\left(0.2\right)\left(1-0.2\right)}\\ =1.386\end{array}$

Interpretation:

The total number of people that would be detected deceptive by lie detector will vary by 1.386 people in comparison of mean of 2.4 people.