Problem 27 - A First Course in Probability

Question

All the workers at a certain company drive to work and park in the company's lot. The company is interested in estimating the average number of workers in a car. Which of the following methods will enable the company to estimate this quantity? Explain your answer. 1\. Randomly choose \(n\) workers, find out how many were in the cars in which they were driven, and take the average of the \(n\) values. 2\. Randomly choose \(n\) cars in the lot, find out how many were driven in those cars, and take the average of the \(n\) values.

Step-by-Step Solution

Answer

The most appropriate method for estimating the average number of workers in a car is Method 2, which involves randomly choosing \(n\) cars in the lot, finding out how many workers were driven in those cars, and taking the average of the \(n\) values. This method takes into account the possibility of workers sharing a car and provides a more representative sample, resulting in a more accurate estimation. Method 1 is not suitable, as it may double-count workers who share a car and doesn't provide an accurate representation of all workers.

Step 1: Method 1 Analysis

Randomly choose \(n\) workers, find out how many were in the cars in which they were driven, and take the average of the \(n\) values. This method attempts to estimate the average number of workers in a car by selecting a sample of workers and investigating their carpooling habits. However, this method would result in double-counting workers who share a car and give different average values depending on which workers are chosen, regardless of the carpooling habits. The average could be higher or lower than the true value as it doesn't account for the possibility that some workers share a car.

Step 2: Method 2 Analysis

Randomly choose \(n\) cars in the lot, find out how many were driven in those cars, and take the average of the \(n\) values. This method attempts to estimate the average number of workers in a car by selecting a sample of cars and counting the number of workers that drove to work in each car. By calculating the average for these values, this method takes into account the fact that some workers may share a car. The randomness of the car selection will help to provide a more representative sample of the carpooling habits of the workers, resulting in a more accurate estimation of the average number of workers in a car.

Step 3: Best method

Method 2 is the most appropriate for estimating the average number of workers in a car, as it considers the possibility of workers sharing a car and provides a more representative sample of the workers' carpooling habits, which will give a more accurate average value. Method 1 is not suitable, as it may double-count workers who share a car and isn't an accurate representation of all workers.