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 $18,000 and the standard deviation is $1000. use the 68 - 95 - 99.7 rule to find what percentage of buyers paid between $18,000 and $20,000. the percentage of buyers who paid between $18,000 and $20,000 is %. (type an exact answer.)

Explanation:

Step1: Recall the 68 - 95 - 99.7 Rule

The 68 - 95 - 99.7 Rule states that in a normal distribution, about 68% of the data lies within 1 standard - deviation of the mean ($\mu\pm\sigma$), about 95% lies within 2 standard - deviations of the mean ($\mu\pm2\sigma$), and about 99.7% lies within 3 standard - deviations of the mean ($\mu\pm3\sigma$). The mean $\mu = 18000$ and the standard deviation $\sigma=1000$.

Step2: Determine the number of standard - deviations from the mean

The value $x = 20000$, and the mean $\mu = 18000$. The number of standard - deviations $z=\frac{x - \mu}{\sigma}=\frac{20000 - 18000}{1000}=2$. The value $x = 18000$ is the mean.

Step3: Use the symmetry of the normal distribution

Since the normal distribution is symmetric about the mean, the percentage of data between the mean ($\mu = 18000$) and $\mu + 2\sigma(=20000)$ is half of the percentage of data between $\mu-2\sigma$ and $\mu + 2\sigma$. The percentage of data between $\mu-2\sigma$ and $\mu + 2\sigma$ is 95%. So the percentage of data between the mean and $\mu + 2\sigma$ is $\frac{95\%}{2}=47.5\%$.

Answer:

47.5