Question

the figure illustrates a normal distribution for the prices paid for a particular model of a new car. the mean is $15,000 and the standard deviation is $500. use the 68 - 95 - 99.7 rule to find the percentage of buyers who paid between $13,500 and $15,000. what percentage of buyers paid between $13,500 and $15,000? %

Explanation:

Step1: Calculate the number of standard deviations

The mean is \(\mu = 15000\) and the standard deviation is \(\sigma=500\).
For \(x = 13500\), the number of standard deviations from the mean is \(z=\frac{15000 - 13500}{500}=\frac{1500}{500}=3\)

Step2: Use the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule states that approximately 99.7% of the data lies within \(z=- 3\) and \(z = 3\) of the mean in a normal distribution.
The distribution is symmetric about the mean. The percentage of data from \(z=-3\) to \(z = 0\) (i.e., from \(13500\) to \(15000\)) is \(\frac{99.7\%}{2}=49.85\%\)

Answer:

\(49.85\)