Question

Suppose a subdivision on the southwest side of Denver, Colorado, contains 1,400 houses. The subdivision was built in 1983. A sample of 110 houses is selected randomly and evaluated by an appraiser. If the mean appraised value of a house in this subdivision for all houses is $227,000, with a standard deviation of $8,500, what is the probability that the sample average is greater than $228,500?

Answer 1

3.22% probability that the sample average is greater than $228,500

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

`\mu = 227000, \sigma = 8500, n = 110, s = (8500)/(√(110)) = 810.44`

What is the probability that the sample average is greater than $228,500?

This is 1 subtracted by the pvalue of Z when X = 228500. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (228500 - 227000)/(810.44)`

`Z = 1.85`

`Z = 1.85` has a pvalue of 0.9678

1 - 0.9678 = 0.0322

3.22% probability that the sample average is greater than $228,500