Question

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: Calculate the number of standard deviations

The mean $\mu = 23000$ and the standard deviation $\sigma=1000$. We want to find the number of standard deviations from the mean for $x = 26000$.
The formula for the number of standard deviations $z=\frac{x-\mu}{\sigma}$.
Substitute $x = 26000$, $\mu = 23000$ and $\sigma = 1000$ into the formula:
$z=\frac{26000 - 23000}{1000}=\frac{3000}{1000}=3$

Step2: Apply the Empirical Rule

The Empirical Rule states that for a normal distribution:

• 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 total percentage of data within $3$ standard deviations of the mean ($\mu - 3\sigma$ to $\mu+3\sigma$) is $99.7\%$. Since the normal distribution is symmetric about the mean ($\mu$), the percentage of data from $\mu$ to $\mu + 3\sigma$ is $\frac{99.7\%}{2}=49.85\%$

Answer:

$49.85\%$