the figure illustrates a normal distribution for the prices paid for a particular model of a new car. the…

the figure illustrates a normal distribution for the prices paid for a particular model of a new car. the mean is $15,000 and the standard deviation is $500. use the 68 - 95 - 99.7 rule to find the percentage of buyers who paid between $13,500 and $15,000. what percentage of buyers paid between $13,500 and $15,000? %

the figure illustrates a normal distribution for the prices paid for a particular model of a new car. the mean is $15,000 and the standard deviation is $500. use the 68 - 95 - 99.7 rule to find the percentage of buyers who paid between $13,500 and $15,000. what percentage of buyers paid between $13,500 and $15,000? %

Answer

Explanation:

Step1: Calculate number of standard - deviations

The mean $\mu = 15000$ and the standard deviation $\sigma=500$. The value $x = 13500$. The number of standard - deviations from the mean is $z=\frac{\mu - x}{\sigma}=\frac{15000 - 13500}{500}=\frac{1500}{500}=3$.

Step2: Apply the 68 - 95 - 99.7 Rule

The 68 - 95 - 99.7 Rule states that about 99.7% of the data lies within 3 standard - deviations of the mean, i.e., between $\mu - 3\sigma$ and $\mu+3\sigma$. The normal distribution is symmetric about the mean. The percentage of data between $\mu - 3\sigma$ and $\mu$ is half of the percentage of data between $\mu - 3\sigma$ and $\mu + 3\sigma$. Since the percentage of data between $\mu - 3\sigma$ and $\mu+3\sigma$ is 99.7%, the percentage of data between $\mu - 3\sigma$ and $\mu$ is $\frac{99.7%}{2}=49.85%$.

Answer:

49.85