Question

Suppose that you are thinking of buying a resale home in a large tract. The owner is asking \(205,500. Your realtor obtains the sale prices of comparable homes in the area that have sold recently. The mean of the prices is \)220,258 and the standard deviation is $5,237. Does it appear that the home you are contemplating buying is a bargain? Explain your answer using the z-score and Chebyshev's rule.

Short answer

Using the z-score and Chebyshev's rule at least 75% of the sale prices should be between$209,784and $230,732 the observations are within the two standard deviations. As the selling price is between the given two results.

Step-by-step solution

Step 1. Given information.

Consider the given question,

The mean of the prices is $220,258 and the standard deviation is $5,237.

Step 2. Verify whether buying is a bargain or not.

Assume the random variable X is defined as the sale prices of home.

Assume the mean sale price of homes in that area $\mu =\$220,258$.

The standard normal score is given below,

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{205258-220258}{5237} \\ =-2.86424 \end{gathered}$$

Here, the standard score of -2.86424 is negative and is observed between -3 and 3 and is not near to 0, so the selling price is not near the mean selling price of homes at that area. Therefore, selling is a bargain.

Step 3. Consider the Chebyshev's rule.

According to Chebyshev's rule, 89%, 75% of the observations are within the three, two standard deviations of the mean.

The standard deviations limits are given below,

Therefore, at least 75% of the sale prices should be between$209,784and $230,732 the observations are within the two standard deviations. As the selling price is between the given two results.