Question

A real estate agent has surveyed houses in twenty nearby zip codes in an attempt to put together a comparison for a new property that she would like to put on the market. The 1057 houses she surveyed have a mean price of ​$169 comma 400​, with a standard deviation of $ 68 comma 438. The mean living area is 2058 sq​ ft, with a standard deviation of 790 sq ft. Which is more​ unusual, a house in that market that sells for ​$300 comma 000 or a house that has 3000 sq ft of living​ area? Explain. Choose the correct answer below. The house that sells for ​$300 comma 000 has a​ z-score of nothing and the house with 3000 sq ft has a​ z-score of nothing. ​Thus, a house in that market that ▼ sells for $ 300 comma 000 has 3000 sq ft of living area is more unusual than a house that ▼ sells for $ 300 comma 000 has 3000 sq ft of living area .

Answer 1

Answer:

A a house in that market that sells for ​$300,000 is unusual.

Step-by-step explanation:

Let the random variable X denote the price of a house and the random variable Y denote the living area of a house.

The number of houses surveyed by the real estate agent is, n = 1057.

Assume that both the random  variables, X and Y are approximately normally distributed.

That is,

$X\sim N(\ 169400,\ \ 68438)\\\\Y\sim N(2058\ \text{sq. ft.},\ 790\ \text{sq. ft.})$

To compute the probability of a Normal distribution we first need to convert the raw scores to z-scores.

$z=\frac{\text{Raw score}-\mu}{\sigma}$

A z-score higher than 1.96 and lower than -1.96 are considered unusual. The values having these z-scores are considered as outliers.

(1)

Compute the z-score for X = 300000 as follows:

$z=\frac{X-\mu}{\sigma}\\\\=\frac{300000-169400}{68438}\\\\=2.70$

(2)

Compute the z-score for Y = 3000 as follows:

$z=\frac{Y-\mu}{\sigma}\\\\=\frac{3000-2058}{790}\\\\=1.19$

The z-score for a house in that market that sells for ​$300,000 is more than 1.96.

This implies, that the price $300,000 is unusually high.

The complete statement is:

The house that sells for ​$300 comma 000 has a​ z-score of 2.70 and the house with 3000 sq ft has a​ z-score of 1.19.