Question

A delivery truck manager takes a sample of 25 delivery trucks and calculates the sample mean and sample standard deviation for the cost of operation. A 95% confidence interval for the population mean cost is constructed and found to be $253 to $320. He reasons that this interval contains the mean operating cost for the entire fleet of delivery trucks since the sample mean is contained in this interval.

1. Do you agree with his reasoning?
2. How would you interpret this confidence interval?
3. Is this an appropriate use of a confidence interval?
4. Concerning your answers to parts a through c, what assumptions did you make (if any)? Does the Central Limit Theorem apply? Why or why not?
5. How does this apply or could apply to your profession?

Answer 1

1. No. The sample mean is always contained within the confidence interval (it is the midpoint) regardless of significance!

2. The interval says that with 95% confidence, the true value of mean operating cost lies between 253 and 320. In other words, there is a 95% chance that the interval contains the true mean operating cost.

3. No. Usually, one uses such reasoning for a difference in means or proportions. If the interval does not contain 0 (ie, there is a difference), then the means are signficantly differently.

4. We don't need to assume the central limit theorem, because the manager could have used the appropriate t-distribution for the small sample size of 25. This would assume the data is normal though. So in the absence of normal data, the analysis is inappropriate, because the sample size needs to be greater than 30 to apply the central limit theorem.