According to the National Association of Realtors, the mean sale price for existing homes in the United States in 2011 was $214,300. Assume that sale prices are normally distributed with a standard deviation of $41,000. Find the percentage of existing homes in 2011 that’s sold for between $91,300 and $214,300.

1. 2.35%
2. 49.85%
3. 99.85%
4. 97.5%

Explanation:

mu = 214300 = population mean of home values

sigma = 41000 = population standard deviation of home values

Compute the z score for the raw score x = 91300

z = (x-mu)/sigma

z = (91300-214300)/41000

z = -123000/41000

z = -3

This tells us the raw score x = 91300 is exactly 3 standard deviations below the mean.

Compute the z score for the raw score x = 214300

z = (x-mu)/sigma

z = (214300-214300)/41000

z = 0/41000

z = 0

This is expected as the raw score x = 214300 is exactly at the mean

Finding P(91300 < x < 214300) is equivalent to computing P(-3 < z < 0)

Recall that the empirical rule states roughly 99.7% of the distribution is within 3 standard deviations of the mean.

In terms of an equation, this means P(-3 < Z < 3) = 0.997 approximately.

Cut 99.7% in half to get (99.7%)/2 = 49.85%

So, P(-3 < Z < 0) = 0.4985 approximately.

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