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.
answer: %

Explanation:

Step1: Calculate the number of standard deviations

The mean $\mu = 23000$, the value $x = 26000$, and the standard deviation $\sigma=1000$.
The number of standard deviations $z=\frac{x - \mu}{\sigma}=\frac{26000 - 23000}{1000}=3$

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 ($\mu\pm3\sigma$)

Since the normal distribution is symmetric about the mean, the percentage of data from the mean ($\mu = 23000$) to $\mu + 3\sigma=23000 + 3\times1000=26000$ is $\frac{99.7\%}{2}=49.85\%$

Answer:

$49.85$