Chapter 7.3, Problem 65E

(a)

To determine

To Calculate: the probability that the mean loss from fire is greater than $300 for an SRS of 1000 homeowners.

Answer to Problem 65E

37.45%

Explanation of Solution

Given:

  $\begin{array}{l}\mu =250\\ \sigma =5000\\ n=1000\\ \overline{x}=300\end{array}$

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

  $z=\frac{x-{\mu }_{\overline{x}}}{{\sigma }_{\overline{x}}}=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}=\frac{300-250}{5000/\sqrt{1000}}=0.32$

The associating probability using the normal probability

  $P\left(Z<0.32\right)$ is given in the row starting with $0.3$ and in the column starting with $0.2$ of the standard normal probability

  $\begin{array}{c}P\left(\overline{X}\ge 300\right)=P\left(Z>0.32\right)\\ =1-P\left(Z<0.32\right)\\ =1-0.6255\\ =0.3745\\ =37.45\%\end{array}$

(b)

To determine

To find: the company charge.

Answer to Problem 65E

$452.39

Explanation of Solution

Given:

  $\begin{array}{c}\mu =250\\ \sigma =5000\\ n=1000\\ P\left(\overline{X}\le \overline{x}\right)=90\%\end{array}$

Formula used:

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

Calculation:

Finding the z-score that associates with a probability of $90\%$ or $0.90$ in the normal probability and it is observed that the closest probability is $0.8997$ which lies in the row $1.2$ and in the column $.08$ of the normal probability table and therefore the corresponding z-score is then $1.2+.08=1.28$

  $z=1.28$

The z-score is

  $z=\frac{x-{\mu }_{\overline{x}}}{{\sigma }_{\overline{x}}}=\frac{\overline{x}-\mu }{\sigma /\sqrt{n}}=\frac{\overline{x}-250}{5000/\sqrt{1000}}$

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

  $\frac{\overline{x}-250}{5000/\sqrt{1000}}=1.28$

  $\overline{x}-250=1.28\left(5000/\sqrt{1000}\right)$

Add 250 to each side:

  $\overline{x}-250+1.28\left(5000/\sqrt{1000}\right)$

Evaluate:

  $x=452.39$

Therefore the company should charge $ $452.39$ .