Chapter 6.2, Problem 67E

To determine

Probability for Lamar and Hareesh finish their jobs within 5 minutes of each other on a randomly selected day.

Answer to Problem 67E

Probability that Lamar and Hareesh finish their jobs within 5 minutes is 0.2048.

Explanation of Solution

Given information:

For L :

Mean,

  ${\mu }_L=105\text{minutes}$

Standard deviation,

  ${\sigma }_L=10\text{minutes}$

For H :

Mean,

  ${\mu }_H=98\text{minutes}$

Standard deviation,

  ${\sigma }_H=15\text{minutes}$

Time taken,

  $x=L-H=-5\text{or}5$ (Within 5 minutes)

Calculations:

The time L it takes Lamar to mow the lawn.

The time H it takes Hareesh to use the edge and string trimmer.

Then

L - H is the difference in time it takes each individual to complete their job.

L and H are independent.

Since L and H have a Normal distribution, their difference ( L - H ) also has a Normal distribution.

Now,

For any two variables,

The mean of the difference is equal to the difference of their means.

  ${\mu }_{L-H}={\mu }_L-{\mu }_H=105-98=7\text{minutes}$

When the random variables are independent, the variance of the difference is equal to the sum of their variances.

  ${\sigma }_{L-H}^2={\sigma }_L^2+{\sigma }_H^2={\left(10\right)}^2+{\left(15\right)}^2=100+225=325\text{minutes}$

We know that

The standard deviation is the square root of the variance.

  ${\sigma }_{L-H}=\sqrt{{\sigma }_{L-H}^2}=\sqrt{325}=5\sqrt{13}\approx 18.0278$

Thus,

The difference ( L - H ) follows a Normal distribution with mean of 7 minutes and standard deviation of approx. 18.0278 minutes.

Now,

For Probability:

Calculate the z − score,

  $z=\frac{x-\mu }{\sigma }=\frac{-5-7}{18.0278}\approx -0.67$

Or

  $z=\frac{x-\mu }{\sigma }=\frac{5-7}{18.0278}\approx -0.11$

Use normal probability table in the appendix, to find the corresponding probability.

See the row that starts with -0.6 and the column that starts with .07 of the standard normal probability table for $P\left(z<-0.67\right)$ .

Or

See the row that starts with -0.1 and the column that starts with .01 of the standard normal probability table for $P\left(z<-0.11\right)$ .

  $\begin{array}{l}P\left(-5<L-H<5\right)=P\left(-0.67<z<-0.11\right)\\ =P\left(z<-0.11\right)-P\left(z<-0.67\right)\\ =0.4562-0.2514\\ =0.2048\end{array}$

Thus,

Probability that Lamar and Hareesh finish their jobs within 5 minutes of each other on a randomly selected day is 0.2048.