Question

The dot plot gives the sale prices for $40$ houses in Ames, Iowa, sold during a recent month. The mean sale price was $\backslash (203,388$ with a standard deviation of $\backslash )87,609$

a. Find the percentile of the house indicated in red on the dot plot.

b. Calculate and interpret the standardized score ($z-$score) for the house indicated by the red dot, which sold for $\$234,000$

Short answer

Part (a) ${65}^{th}$percentile

Part (b) The house's sales price of $\$234000$ is $0.35$ standard deviations more than the mean sold price of $\$203388$

Step-by-step solution

Part (a) Step 1: Given information

Part (a) Step 2: Concept

$percentile=\frac{numberofdatavaluesthatarelessthantheindividual\text{'}sdatavalue}{totalnumberofdatavalues}\times 100\%$

Part (a) Step 3: Calculation

The dot plot shows that $26$ of the $40$ dots to the left of the red dot have a price below $\$234000$ which is the certain value linked with the red dot. This means that $26$ of the $40$ dwellings in the sample have a price below $\$234000$ The percentile is calculated by dividing the number of data values less than the individual's data value by the total number of data values.

$percentile=\frac{numberofdatavaluesthatarelessthantheindividual\text{'}sdatavalue}{totalnumberofdatavalues}\times 100\%$

$$\begin{gathered} =\frac{26}{40}\times 100 \\ =65\% \end{gathered}$$

As a result, the percentile associated with the red dot is the ${65}^{th}$ percentile.

Part (b) Step 1: Calculation

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{234000-203388}{87609} \\ =0.35 \end{gathered}$$

The $z-$score displays how many standard deviations a value deviates from the mean, with a negative $z-$score indicating a value below the mean and a positive $z-$score indicating a value above the mean.

The house's sales price of $\$234000$ is $0.35$standard deviations more than the mean selling price of $\$203388$