2. Interval estimation basics - Interpretation of confidence intervals

Interval estimation is the process of using a sample statistic to estimate a range of values in which the population parameter falls. The confidence levelis the level of confidence associated with the confidence interval estimate. The confidence level is determined by the alpha, which is the risk you are willing to take of being wrong.

Suppose your confidence level is 91%. Assuming you employ repeated random sampling and compute the confidence interval estimate for each sample, you would expect that of the intervals the population value.

Historically, a typical home in the U.S. costs 2.5 times the typical annual income. For example, if the median income in a geographic area is $100,000, the median price of a home in that area would be 2.5 x $100,000 = $250,000. Unfortunately, since 1999, wage increases have not kept up with home prices, particularly in large urban areas. Homeowners were paying up to 52% more for homes at the end of 2012 relative to the median income in those areas.

Suppose you take a random sample of 1,000 homes in Denver. You find that the average home price in your sample has increased 41.3% with a standard deviation of s = 2.1% since 1999. You want to estimate the average home increase in Denver with a fairly small risk of error, so you set alpha as 0.0001.

This means you can be confident that increase in home prices in Denver will be the confidence interval appropriate for this alphain this case, 41.3% 0.3%.

Suppose that the median price for a two-bedroom house in Denver is $271,175 and the average income for a person living there is $57,000. This house price is definitely more than 2.5 times the income. Given your other expenses (such as student loan debt, etc.), would you be able to afford to buy a two-bedroom home in Denver?