Chapter 7.3, Problem 66E

(a)

To determine

To Calculate: the probability that an SRS of 40 customers grab an average of $515 or more.

Answer to Problem 66E

7.93%

Explanation of Solution

Given:

  $\begin{array}{c}\mu =13\\ \sigma =9\\ n=40\\ \overline{x}=15\end{array}$

Formula used:

  $z=\frac{x-{\mu }_{\overline{x}}}{{\sigma }_{\overline{x}}}$

Calculation:

Since the population distribution is normal, the sampling distribution of the sample mean $\overline{x}$ is also normal.

The Z-score is

  $z=\frac{x-{\mu }_{\overline{x}}}{{\sigma }_{\overline{x}}}=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}=\frac{15-13}{9/\sqrt{40}}=1.41$

The associating probability using the normal probability

  $P\left(Z<1.41\right)$ is given in the row starting with $1.4$ and in the column starting with $.01$ of the standard normal probability table in the appendix.

  $\begin{array}{c}P\left(\overline{X}\ge 15\right)=P\left(Z>1.41\right)\\ =1-P\left(Z<1.41\right)\\ =1-0.9207\\ =0.0793\\ =7.93\%\end{array}$

(b)

To determine

To find: the charge to play the game.

Answer to Problem 66E

$15.37

Explanation of Solution

Given:

  $\begin{array}{c}\mu =13\\ \sigma =9\\ n=40\\ P\left(\overline{X}\le \overline{x}\right)=95\%\end{array}$

Formula used:

  $z=\frac{x-{\mu }_{\overline{x}}}{{\sigma }_{\overline{x}}}$

Calculation:

Finding the z-score that associates with a probability of $95\%$ or $0.95$ in the normal probability table and The probability $0.95$ lies exactly between 0.9495 and 0.9505, where associates with the z-scores 1.64 and 1.64 it can use the z-score exactly in the middle between $1.64\text{ and 1}\text{.65}\text{.}$ to represent the probability 0.95 ,which is $1.645$ .

  $z=1.645$

The z-score is

  $z=\frac{x-{\mu }_{\overline{x}}}{{\sigma }_{\overline{x}}}=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}=\frac{\overline{x}-13}{9/\sqrt{40}}$

The two found expressions of the z-score than have to be equal:

  $\frac{\overline{x}-13}{9/\sqrt{40}}=1.645$

Multiply each side by $9/\sqrt{40}$ :

  $\overline{x}-13=1.645\left(9/\sqrt{40}\right)$

Add 13 to each side:

  $\overline{x}+13=1.645\left(9/\sqrt{40}\right)$

Evaluate:

  $x=15.34$

Thus the company should charge $15.34