Basic Business Statistics Concepts and Applications by Mark Berenson, David Levine, Kathryn Szabat, David Stephan (Textbook Question)

Question

problem 13.61) In Problem 13.9 on page 494, an agent for a real estate company wanted to predict the monthly rent for one-bedroom apartments, based on the size of an apartment. The data are stored in RentSilverSpring .

a. Construct a 95% confidence interval estimate of the mean monthly rental for all one-bedroom apartments that are 800 square feet in size.

b. Construct a 95% prediction interval of the monthly rental for an individual one-bedroom apartment that is 800 square feet in size.

c. Explain the difference in the results in (a) and (b).

Answer 1

I don't have access to this textbook, but I can still provide a general explanation of the difference between a confidence interval and a prediction interval, which may help you understand the concepts related to parts (a) and (b) of the problem:

a. Confidence interval: A confidence interval is used to estimate a population parameter (such as the mean) based on a sample statistic. In this case, the agent wants to estimate the mean monthly rental for all one-bedroom apartments that are 800 square feet in size. By constructing a 95% confidence interval, the agent will have an interval estimate within which they can be reasonably confident that the true population mean lies. This interval provides a range of values within which the mean monthly rental is expected to fall, with a certain level of confidence.

b. Prediction interval: A prediction interval is used to estimate the range within which an individual observation is expected to fall. In this case, the agent wants to predict the monthly rental for an individual one-bedroom apartment that is 800 square feet in size. By constructing a 95% prediction interval, the agent will have an interval estimate within which they can be reasonably confident that the monthly rental for a specific apartment lies. This interval takes into account the variability in the data and provides a range of values within which the actual monthly rental for an individual apartment is expected to fall, with a certain level of confidence.

The key difference between a confidence interval and a prediction interval is in their interpretation and the scope of inference. A confidence interval is used to estimate the population parameter (mean in this case) based on a sample, while a prediction interval is used to estimate the range of values for an individual observation. The prediction interval is typically wider than the confidence interval since it includes both the uncertainty in estimating the mean and the inherent variability of individual observations.