Question

the following dot - plot gives the sale prices for 40 houses in ames, iowa, sold during a recent month. the mean sale price was $213,388 with a standard deviation of $87,609. (a) find the percentile of the house represented by the red dot. (enter an integer.) (b) calculate and interpret the standardized score (z - score) for the house represented by the red dot, which sold for $234,000. z = - 0.35. this home has a sale price that is $350 below the mean sale price. z = - 0.35. this home has a sale price that is 0.35 standard deviations below the mean sale price. z = 0.35. this home has a sale price that is $350 above the mean sale price. z = 0.35. this home has a sale price that is 0.35 standard deviations above the mean sale price.

Explanation:

Step1: Use z - table for percentile

We know that for a z - score, we can use the standard normal distribution table (z - table). The z - score of the red - dot house is \(z=- 0.35\). Looking up the value in the standard normal table for \(z = - 0.35\), the area to the left of \(z=-0.35\) is approximately \(0.3632\).

Step2: Convert to percentile

Percentile is the percentage of data values that are less than or equal to a particular value. So the percentile is approximately \(36\) (rounding \(0.3632\times100\) to the nearest integer).

Step3: Interpret z - score

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the data value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. A z - score of \(z = - 0.35\) means that the value of \(x\) (the sale price of the house) is \(0.35\) standard deviations below the mean \(\mu\).

Answer:

(a) 36
(b) \(z=-0.35\). This home has a sale price that is \(0.35\) standard deviations below the mean sale price.