Chapter SH, Problem 11.11EP

a.

To determine

To find probability that there will be more than $10$ assemblies in a given year.

Answer to Problem 11.11EP

  $0.93319$

Explanation of Solution

Given:

The number of assemblies at West high school each year is normally distributed with a mean of $12.4$ and a standard deviation of $1.6$ .

Calculation:

  $\mu =12.4$ , $\sigma =1.6$

The probability that there will be more than $10$ assemblies in a given year will be represented by. Compute the corresponding z-value for $X=10$ as follow:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{10-12.4}{1.6}\\ =\frac{-2.4}{1.6}\\ =-1.5\end{array}$

Hence, the required probability will be given below:

  $\begin{array}{l}P(X>10)=P(z>-1.5)\\ =1-P(z\le -1.5)\\ =1-0.06681\\ =0.93319\end{array}$

Hence, the required probability is $0.93319$ .

b.

To determine

To find the number of years that there were between $11$ and $13$ assemblies in a given year

Answer to Problem 11.11EP

  $14$ Years.

Explanation of Solution

Given:

The number of assemblies at West high school each year is normally distributed with a mean of $12.4$ and a standard deviation of $1.6$ .

Calculation:

The probability that there were between $11$ and $13$ assemblies in a given year will be represented by $P(11\le X\le 13)$ . Compute the corresponding z-value for $X=11$ as follow:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{11-12.4}{1.6}\\ =\frac{-1.4}{1.6}\\ =-0.875\end{array}$

Compute the corresponding z-value for $X=13$ as follow:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{13-12.4}{1.6}\\ =\frac{0.6}{1.6}\\ =0.375\end{array}$

Hence, the required probability will be calculated below:

  $\begin{array}{l}P(11\le X\le 13)=P(-0.875\le z\le 0.375)\\ =P(z\le 0.375)-P(z\le -0.875)\\ =0.64617-0.19079\\ =0.45538\end{array}$

If the school is existed for $30$ years then the number of years in which between $11$ and $13$ assemblies conducted, is calculated below:

  $\begin{array}{l}30\times 0.45538\\ \approx 14\end{array}$

Hence, about $14$ years in which there were between $11$ and $13$ assemblies.