Question

not everyone pays the same price for the same model of a car. the figure illustrates a normal distribution for the prices paid for a particular model of a new car. the mean is $19,000 and the standard deviation is $1000. use the 68 - 95 - 99.7 rule to find what percentage of buyers paid between $16,000 and $19,000. the percentage of buyers who paid between $16,000 and $19,000 is □%. (type an exact answer.)

Explanation:

Step1: Calculate the number of standard deviations from the mean

The mean is \(\mu = 19000\) and the standard deviation is \(\sigma=1000\).
For \(x = 16000\), the number of standard deviations \(z=\frac{\mu - x}{\sigma}=\frac{19000 - 16000}{1000}=3\)

Step2: Use the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule states that:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation of the mean (\(\mu\pm\sigma\))
• Approximately \(95\%\) of the data lies within \(2\) standard deviations of the mean (\(\mu\pm2\sigma\))
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations of the mean (\(\mu\pm3\sigma\))

The interval from \(\mu - 3\sigma\) to \(\mu\) is half of the interval from \(\mu - 3\sigma\) to \(\mu+3\sigma\).

Since the normal distribution is symmetric about the mean, the percentage of data from \(\mu - 3\sigma\) to \(\mu\) is \(\frac{99.7\%}{2}=49.85\%\)

Answer:

\(49.85\)