Chapter 7, Problem 1SR

Microwave ovens only last so long. The life-time of a microwave oven follows a uniform distribution between 8 and 14 years.

1. (a) Draw this uniform distribution. What are the height and base values?
2. (b) Show the total area under the curve is 1.00.
3. (c) Calculate the mean and the standard deviation of this distribution.
4. (d) What is the probability a particular microwave oven lasts between 10 and 14 years?
5. (e) What is the probability a microwave oven will last less than 9 years?

To determine

Draw the uniform distribution graphically.

Find the height and base values of distribution.

Answer to Problem 1SR

The height and base values of the distribution are 0.167 and 6, respectively.

Explanation of Solution

Step-by-step procedure to obtain the uniform distribution using MINITAB software:

• Choose Graph > Probability Distribution Plot.
• From Distribution, choose Uniform.
• Enter Lower endpoint as 8 and Upper endpoint as 14.
• Click Ok.

The output obtained using MINITAB software is represented as follows:

From the above output, the shape of the distribution is rectangle.

The height of the distribution is calculated below:

$\begin{array}{c}\text{Height}=\frac{1}{b-a}\\ =\frac{1}{14-8}\\ =\frac{1}{6}\end{array}$

Therefore, the height of the distribution is 0.167.

The base of the distribution is obtained below:

$\begin{array}{c}\text{Base}=b-a\\ =14-8\\ =6\end{array}$

The base value of the distribution is 6.

To determine

Prove that the total area under the curve is 1.00

Explanation of Solution

Let X is the life-time of a microwave oven which follows uniform distribution over the interval from 8 and 14 years.

That is, $a=8$ and $b=14$.

The probability density function of a uniform distribution is,

$P\left(X\right)=\{\begin{array}{l}\frac{1}{b-a}\text{ ;a}\le x\le b\\ 0\text{ ;elsewhere}\text{.}\end{array}$

The height and base values of the distribution are 0.167 and 6 respectively.

The area under the curve is obtained below:

$\begin{array}{c}\text{Area=}\left(\text{height}\right)\left(\text{length}\right)\\ =\frac{1}{6}\times 6\\ =1\end{array}$

Therefore, the total area under the curve is 1.

To determine

Compute the mean and standard deviation of the distribution.

Answer to Problem 1SR

The mean of the distribution is 11.

The standard deviation of the distribution is 1.73.

Explanation of Solution

The formula for mean of the distribution is stated below:

$\begin{array}{c}\mu =\frac{a+b}{2}\\ =\frac{8+14}{2}\\ =\frac{22}{2}\\ =11\end{array}$

The mean life-time of a microwave oven is 11.

The formula for standard deviation of the distribution is computed below:

$\begin{array}{c}\sigma =\sqrt{\frac{{\left(b-a\right)}^2}{12}}\\ =\sqrt{\frac{{\left(14-8\right)}^2}{12}}\\ =\sqrt{\frac{36}{12}}\\ =\sqrt{3}\\ =1.73\end{array}$

Therefore, the standard deviation of the distribution is 1.73.

To determine

Find the probability that a particular microwave oven lasts between 10 and 14 years.

Answer to Problem 1SR

The probability that a particular microwave oven lasts between 10 and 14 years is 0.668.

Explanation of Solution

The height of the distribution is 0.167 and base of the distribution is 4 ( = 14 – 10).

The probability that a particular microwave oven lasts between 10 and 14 years is,

$\begin{array}{c}P\left(10<X<14\right)=\left(\text{height}\right)\left(\text{base}\right)\\ =\left(0.167\right)\left(4\right)\\ =0.668\end{array}$

Therefore, the probability that a particular microwave oven lasts between 10 and 14 years is 0.668.

To determine

Find the probability that a microwave oven will last less than 9 years.

Answer to Problem 1SR

The probability that a particular microwave oven will last less than 9 years is 0.167.

Explanation of Solution

The height of the distribution is 0.167 and base of the distribution is 1.

The probability that a particular microwave oven will last less than 9 years is calculated below:

$\begin{array}{c}P\left(8<X<9\right)=\left(\text{height}\right)\left(\text{base}\right)\\ =\left(0.167\right)\left(1\right)\\ =0.167\end{array}$

Therefore, the probability that a particular microwave oven will last less than 9 years is 0.167.