Question

5) The National Association of Realtors estimatest that 23% of all homes purchased in 2004 were considered

investment properties. If a random sample of 800 homes sold in 2004 is obtained, what is the probability that al
most 25% of those are used as investment property?

Answer 1

0.9099 = 90.99% probability that at most 25% of those are used as investment property.

Explanation:

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

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, 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.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

23% of all homes purchased in 2004 were considered investment properties.

This means that
`p = 0.23`

Sample of 800 homes

This means that
`n = 800`

Mean and Standard deviation:

`\mu = p = 0.23`

`\sigma = \sqrt{(p(1-p))/(n)} = \sqrt{(0.23*0.77)/(800)} = 0.0149`

What is the probability that at most 25% of those are used as investment property?

This is the pvalue of Z when X = 0.25. So

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

`Z = (0.25 - 0.23)/(0.0149)`

`Z = 1.34`

`Z = 1.34` has a pvalue of 0.9099

0.9099 = 90.99% probability that at most 25% of those are used as investment property.