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 $14,000 and the standard deviation is $500. use the 68 - 95 - 99.7 rule to find the percentage of buyers who paid between $12,500 and $14,000. what percentage of buyers paid between $12,500 and $14,000? %

the figure illustrates a normal distribution for the prices paid for a particular model of a new car. the mean is $14,000 and the standard deviation is $500. use the 68 - 95 - 99.7 rule to find the percentage of buyers who paid between $12,500 and $14,000. what percentage of buyers paid between $12,500 and $14,000? %

Answer

Explanation:

Step1: Calculate the z - scores

The mean $\mu = 14000$ and the standard deviation $\sigma= 500$. The z - score formula is $z=\frac{x - \mu}{\sigma}$. For $x = 12500$, $z=\frac{12500 - 14000}{500}=\frac{- 1500}{500}=-3$. For $x = 14000$, $z=\frac{14000 - 14000}{500}=0$.

Step2: Apply the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule states that about 68% of the data lies within 1 standard deviation of the mean, about 95% lies within 2 standard deviations, and about 99.7% lies within 3 standard deviations. The area between $z=-3$ and $z = 0$ is half of the area between $z=-3$ and $z = 3$. Since the area between $z=-3$ and $z = 3$ is 99.7%, the area between $z=-3$ and $z = 0$ is $\frac{99.7%}{2}=49.85%$.

Answer:

49.85%