Chapter 7, Problem 102CE

Times for a surgical procedure are normally distributed. There are two methods. Method A has a mean of 28 minutes and a standard deviation of 4 minutes, while method B has a mean of 32 minutes and a standard deviation of 2 minutes. (a) Which procedure is preferred if the procedure must be completed within 28 minutes? (b) Thirty-eight minutes? (c) Thirty-six minutes? Explain your reasoning fully.

To determine

Find the better procedure if the procedure must be completed within 28 minutes.

Answer to Problem 102CE

The better method for John to reach at the airport in 28 minutes to pick up his spouse is method A.

Explanation of Solution

Calculation:

It is given that the time on method A follows normal distribution with mean of 28 minutes, standard deviation of 4 minutes and the time on method B follows normal distribution with mean of 32 minutes, standard deviation of 2 minutes.

Normal distribution:

A continuous random variable X is said to follow normal distribution if the probability density function of X is,

$f\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}},-\infty <x<\infty$, with mean $\mu$ and standard deviation $\sigma$.

Assume that the random variable X denotes the times for a surgical procedure.

For method A:

It is given that ${\mu }_A=28\text{minutes}$ and ${\sigma }_A=4\text{minutes}$.

Now, the probability to complete the procedure within 28 minutes using method A, implies that $P\left(X<28\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(28,28,4,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<28\right)=0.5$.

Thus, the probability to complete the procedure within 28 minutes using method A is 0.5.

For method B:

It is given that ${\mu }_B=32\text{minutes}$ and ${\sigma }_B=2\text{minutes}$.

Now, the probability to complete the procedure within 28 minutes using method B, implies that $P\left(X<28\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(28,32,2,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<54\right)=0.0228$.

Thus, the probability to complete the procedure within 28 minutes using method B is 0.0228.

Therefore, there is to complete the procedure within 28 minutes using method A than by method B.

Thus, the better procedure if the procedure must be completed within 28 minutes is method A.

To determine

Find the better procedure if the procedure must be completed within 38 minutes.

Answer to Problem 102CE

The better method for John to reach at the airport in 38 minutes to pick up his spouse is method B.

Explanation of Solution

Calculation:

For method A:

It is given that ${\mu }_A=28\text{minutes}$ and ${\sigma }_A=4\text{minutes}$.

Now, the probability to complete the procedure within 38 minutes using method A, implies that $P\left(X<38\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(38,28,4,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<38\right)=0.9938$.

Thus, the probability to complete the procedure within 38 minutes using method A is 0.9938.

For method B:

It is given that ${\mu }_B=32\text{minutes}$ and ${\sigma }_B=2\text{minutes}$.

Now, the probability to complete the procedure within 38 minutes using method B, implies that $P\left(X<38\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(38,32,2,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<38\right)=0.9987$.

Thus, the probability to complete the procedure within 38 minutes using method B is 0.9987.

Therefore, there is to complete the procedure within 38 minutes using method B than by method A.

Thus, the better procedure if the procedure must be completed within 38 minutes is method B.

To determine

Find the better procedure if the procedure must be completed within 36 minutes.

Answer to Problem 102CE

The better procedure if the procedure must be completed within 36 minutes is method B.

Explanation of Solution

Calculation:

Coefficient of variation:

Coefficient of variation for a random variable X is defined as,

$\text{CV}=\frac{s}{\overline{x}}\times 100\%$ , where s is the sample standard deviation and $\overline{x}$ is the same mean of X.

It is better to use a random variable with lower CV.

For method A:

It is given that ${\mu }_A=28\text{minutes}$ and ${\sigma }_A=4\text{minutes}$.

Now, the probability to complete the procedure within 36 minutes using method A, implies that $P\left(X<36\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(36,28,4,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<36\right)=0.9772$.

Thus, the probability to complete the procedure within 36 minutes using method A is 0.9772.

The CV for method A is,

$\begin{array}{c}{\text{CV}}_A=\frac{4}{28}\times 100\%\\ =0.1428\times 100\%\\ =14.28\%\end{array}$

For method B:

It is given that ${\mu }_B=32\text{minutes}$ and ${\sigma }_B=2\text{minutes}$.

Now, the probability to complete the procedure within 36 minutes using method B, implies that $P\left(X<36\right)$.

Probability value:

Software procedure:

Step-by-step software procedure to obtain probability value using EXCEL is as follows:

• Open an EXCEL file.
• In cell A1, enter the formula “=NORM.DIST(36,32,2,1)”.
• Output using EXCEL software is given below:

Therefore, $P\left(X<36\right)=0.9772$.

Thus, the probability to complete the procedure within 36 minutes using method B is 0.9772.

The CV for method B is,

$\begin{array}{c}{\text{CV}}_B=\frac{2}{32}\times 100\%\\ =0.0625\times 100\%\\ =6.25\%\end{array}$

Therefore, there is same chance to complete the procedure within 36 minutes by method A and method B.

Now, the coefficient of variation for method B is less than method A.

Thus, the better procedure if the procedure must be completed within 36 minutes is method B.