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 $20,000 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: Calculate the number of standard deviations

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

Step2: Apply the 68 - 95 - 99.7 Rule

The 68 - 95 - 99.7 Rule states that about 95% of the data lies within $\mu\pm2\sigma$.
The interval $(\mu - 2\sigma,\mu+2\sigma)=(20000-2\times1000,20000 + 2\times1000)=(18000,22000)$
The distribution is symmetric about the mean $\mu = 20000$.
The percentage of data between $\mu-2\sigma$ and $\mu$ is half of the percentage of data between $\mu - 2\sigma$ and $\mu+2\sigma$.

Since the percentage of data between $\mu - 2\sigma$ and $\mu+2\sigma$ is 95%, the percentage of data between $\mu-2\sigma$ and $\mu$ is $\frac{95\%}{2}=47.5\%$

Answer:

47.5