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 $17,000 and the standard deviation is $500. use the 68 - 95 - 99.7 rule to find the percentage of buyers who paid between $15,500 and $17,000. what percentage of buyers paid between $15,500 and $17,000? %

the figure illustrates a normal distribution for the prices paid for a particular model of a new car. the mean is $17,000 and the standard deviation is $500. use the 68 - 95 - 99.7 rule to find the percentage of buyers who paid between $15,500 and $17,000. what percentage of buyers paid between $15,500 and $17,000? %

Answer

Explanation:

Step1: Calculate number of standard - deviations

The mean $\mu = 17000$ and the standard deviation $\sigma=500$. The value $x = 15500$. The number of standard - deviations $z=\frac{x - \mu}{\sigma}=\frac{15500 - 17000}{500}=\frac{- 1500}{500}=-3$.

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, about 95% lies within 2 standard - deviations, and about 68% lies within 1 standard - deviation. The 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