Many variables important to the real estate market are skewed, limited to only a few values or considered as categorical variables. Yet marketing and business decisions are often made based on means and proportions calculated over many homes. One reason these statistics are useful is the Central Limit Theorem.

Data on 1063 houses sold recently in the Saratoga, New York, area are in the file ch10_MCSP_Real_Estate. Let’s investigate how the CLT guarantees that the sampling distribution of proportions approaches Normal and that the same is true for means of a quantitative variable even when samples are drawn from populations that are far from Normal.

Part 1: Proportions

The variable Fireplace is a dichotomous variable, where 1 = has a fireplace and 0 = does not have a fireplace.

• Calculate the proportion of homes that have fireplaces for all 1063 homes. Using this value, calculate what the standard error of the sample proportion would be for a sample of size 50.

• Using the software of your choice, draw 100 samples of size 50 from this population of homes, find the proportion of homes with fireplaces in each of these samples, and make a histogram of these proportions.

• Compare the mean and standard deviation of this (sampling) distribution to what you previously calculated.

Part 2: Means

• Select one of the quantitative variables and make a histogram of the entire population of 1063 homes. Describe the distribution (including its mean and SD).

• Using the software of your choice, draw 100 samples of size 50 from this population of homes, find the means of these samples, and make a histogram of these means.

• Compare the (sampling) distribution of the means to the distribution of the population.

• Repeat the exercise with samples of sizes 10 and 30. What do you notice about the effect of the sample size?

Some statistics packages make it easier than others to draw many samples and find means. Your instructor can provide advice on the path to follow for your package. If you’re using Excel, you’ll need to use the DDXL add-in to make your histograms.

An alternative approach is to have each member of the class draw one sample to find the proportion and mean and then combine the statistics for the entire class.