Problem 69 - Applied Statistics and Probability for Engineers

Question

An article in Knee Surgery Sports Traumatology, Arthroscopy ["Effect of Provider Volume on Resource Utilization for Surgical Procedures" (2005, Vol. 13, pp. \(273-279\) ) showed a mean time of 129 minutes and a standard deviation of 14 minutes for ACL reconstruction surgery for high-volume hospitals (with more than 300 such surgeries per year). If a high-volume hospital needs to schedule 10 surgeries, what are the mean and variance of the total time to complete these surgeries? Assume that the times of the surgeries are independent and normally distributed.

Step-by-Step Solution

Answer

Mean: 1290 minutes, Variance: 1960 minutes^2

Step 1: Understanding the Problem

We are given a mean time of 129 minutes and a standard deviation of 14 minutes for a single ACL reconstruction surgery in high-volume hospitals. We need to find the mean and variance of the total time to complete 10 independent surgeries.

Step 2: Calculating the Mean of Total Time

Since the mean time for one surgery is 129 minutes, the mean time for 10 surgeries is simply 10 times the mean time of one surgery. \[ \text{Mean of total time} = 10 \times 129 = 1290 \text{ minutes} \]

Step 3: Calculating the Variance of Total Time

The variance for the time of one surgery is the square of the standard deviation: \[ \text{Variance of one surgery} = 14^2 = 196 \text{ minutes}^2 \] For independent surgeries, the variance of the total time is 10 times the variance of one surgery:\[ \text{Variance of total time} = 10 \times 196 = 1960 \text{ minutes}^2 \]

Key Concepts

mean

In statistics, the mean is an average that provides a central value for a set of numbers. It's an easy way to get a sense of the general trend or the most likely expectation within a given data set. In the context of our exercise, the mean time taken for one ACL reconstruction surgery is given as 129 minutes. This means if you were to pick one surgery at random from this high-volume hospital's records, 129 minutes would be the expected duration.
To find the mean time for 10 surgeries, we multiply the mean time for one surgery by the number of surgeries. Hence, the mean total time is 1290 minutes. This simple multiplication works under the assumption that each surgery time is independent and identically distributed.

variance

Variance provides a measure of how much the data in a set are spread out from the mean. It tells us about the variability or dispersion. In simpler terms, variance indicates how far each number in your data set is from the mean and from every other number.
For a single surgery, we calculate variance as the square of the standard deviation, which is 14 minutes, resulting in a variance of 196 minutes squared. The variance for the total time of 10 surgeries will be 10 times the variance of one surgery, equating to 1960 minutes squared. This calculation assumes that each surgery is an independent event, meaning its time is not influenced by the times of other surgeries.

independent events

In probability and statistics, independent events are those events whose outcomes do not affect each other. Knowing the outcome of one event doesn’t change the likelihood of another.
This concept is crucial for our calculations because it allows certain mathematical simplifications.

• When events are independent, the probability or variance of their occurrences can be calculated as the product of their individual probabilities or variances.
• In our exercise, surgeries are considered independent, hence we can multiply the variance of one surgery's time by the number of surgeries to get the total variance.

Assuming independence simplifies the analysis significantly while analyzing multiple events.

normally distributed

A normally distributed variable is one that follows a bell curve, implying that most of its observations cluster around the mean, and probabilities tail off symmetrically on both sides. It is a common assumption in many statistical models.
For surgeries at a high-volume hospital, the assumption of normal distribution simplifies estimations of probabilities around the mean time. This model suggests that most surgeries will take around the mean time of 129 minutes, with fewer surgeries as the time moves away in either direction.

• This distribution helps in making predictions about future surgeries, understanding luck (deviations from the mean), and planning hospital resources accordingly.

Understanding the properties of a normal distribution can be fundamental in medicine and operational planning, which typically demands precision and efficiency.