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 $2000. use the 68 - 95 - 99.7 rule to find what percentage of buyers paid between $19,000 and $25,000. the percentage of buyers who paid between $19,000 and $25,000 is % (type an exact answer.)

Explanation:

Step1: Calculate the number of standard deviations

The mean is $\mu = 19000$ and the standard deviation is $\sigma=2000$.
For $x = 25000$, the number of standard deviations from the mean is $z=\frac{x-\mu}{\sigma}=\frac{25000 - 19000}{2000}=\frac{6000}{2000}=3$.

Step2: Apply the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 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 area under the normal curve is 100%. The area from $-\infty$ to $\mu$ is 50%. The area from $\mu$ to $\mu + 3\sigma$ is $\frac{99.7\%}{2}=49.85\%$

Answer:

49.85