Question

answer:
not everyone pays the same price for the same model of a car. the figure shows a normal distribution for the prices paid for a
particular model of a new car. the mean is $23,000 and the standard deviation is $1000. use the empirical rule to find the
percentage of buyers who paid between $23,000 and $26,000.

Explanation:

Step1: Recall the Empirical Rule

The Empirical Rule states that for a normal distribution:

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

We are given \(\mu = 23000\) and \(\sigma=1000\)

The interval is from \(x_1 = 23000\) to \(x_2=26000\)

First, find the number of standard deviations from the mean. Use the formula \(z=\frac{x-\mu}{\sigma}\)

For \(x = 26000\), \(z=\frac{26000 - 23000}{1000}=\frac{3000}{1000}=3\)

For \(x = 23000\), \(z=\frac{23000 - 23000}{1000}=0\)

Step2: Use the Empirical Rule

Since the normal distribution is symmetric about the mean (\(z = 0\)). The total area under the normal - distribution curve is \(1\) (or \(100\%\))

The area within \(z=- 3\) and \(z = 3\) is \(99.7\%\) (from the Empirical Rule). The area within \(z=-3\) and \(z = 0\) is the same as the area within \(z = 0\) and \(z = 3\)

So the area within \(z = 0\) and \(z = 3\) is \(\frac{99.7\%}{2}=49.85\%\)

Answer:

\(49.85\%\)