Chapter 6.1, Problem 13E

a.

To determine

To explain: The reason of mean of Y locating at the solid red line.

Explanation of Solution

Given:

The figure is:

  

The provided figure shows that the distribution is symmetric because the mean is the at the center at 5 which implies that the distribution is approximately normally distributed.

b.

To determine

To find: The expected value of the first digit.

Answer to Problem 13E

The expected value is 3.441.

Explanation of Solution

The expected value of the first digit can be computed as:

  $\begin{array}{c}E\left(X\right)={\displaystyle \sum x\times P\left(x\right)}\\ =1\left(0.301\right)+2\left(0.176\right)+\mathrm{.........}+9\left(0.046\right)\\ =3.441\end{array}$

This indicates that the mean of the first digit is 3.441.

c.

To determine

To find: The $P\left(Y>6\right)$ using the uniform distribution and Benford’s law. Also, check if the report is correct or not.

Answer to Problem 13E

Expense report is incorrect.

Explanation of Solution

  $P\left(Y>6\right)$ can be calculated as:

  $\begin{array}{c}P\left(Y>6\right)=P\left(Y=7\right)+P\left(Y=8\right)+P\left(Y=9\right)\\ =\frac{1}{9}+\frac{1}{9}+\frac{1}{9}\\ =0.333\end{array}$

The probability using Benford’s law can be computed as:

  $\begin{array}{c}P\left(Y>6\right)=P\left(Y=7\right)+P\left(Y=8\right)+P\left(Y=9\right)\\ =0.058+0.051+0.046\\ =0.155\end{array}$

Since, the proportion is closer to 0.155. It could be concluded that the report of the expense is incorrect.