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 difference from the mean

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

Step2: Apply the 68 - 95 - 99.7 Rule

The 68 - 95 - 99.7 Rule states that about \(99.7\%\) of the data lies within \(3\) standard deviations of the mean (\(\mu\pm3\sigma\)), about \(95\%\) within \(2\) standard deviations (\(\mu\pm2\sigma\)), and about \(68\%\) within \(1\) standard deviation (\(\mu\pm1\sigma\)).
The interval from \(13500\) (\(\mu - 3\sigma\)) to \(15000\) (\(\mu\)) is half of the interval \(\mu - 3\sigma\) to \(\mu+3\sigma\).

Answer:

\(49.85\%\)