Question

A group of realtors estimates that 23% of all homes purchased last year were considered investment properties.

If a sample of 800 homes sold last year is obtained, what is the probability that at most 200 homes are going to used as investment property?

Answer 1

90.99% probability that at most 200 homes are going to used as investment property

Explanation:

I am going to use the binomial approximation to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Normal probability distribution

Problems of normally distributed samples can be 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.

When we are approximating a binomial distribution to a normal one, we have that
`\mu = E(X)`,
`\sigma = √(V(X))`.

In this problem, we have that:

`p = 0.23, n = 800`

So

`\mu = E(X) = np = 800*0.23 = 184`

`\sigma = √(V(X)) = √(np(1-p)) = √(800*0.23*0.77) = 11.90`

If a sample of 800 homes sold last year is obtained, what is the probability that at most 200 homes are going to used as investment property?

This is the pvalue of Z when X = 200. So

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

`Z = (200 - 184)/(11.90)`

`Z = 1.34`

`Z = 1.34` has a pvalue of 0.9099

90.99% probability that at most 200 homes are going to used as investment property