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 $21,000 and the standard deviation is $1000. use the 68 - 95 - 99.7 rule to find what percentage of buyers paid between $19,000 and $21,000. the percentage of buyers who paid between $19,000 and $21,000 is % (type an exact answer.)

Explanation:

Step1: Recall the 68 - 95 - 99.7 Rule

The 68 - 95 - 99.7 Rule states that for a normal distribution:

• Approximately 68% of the data lies within \( \mu\pm\sigma\) (one standard deviation of the mean),
• Approximately 95% of the data lies within \( \mu\pm2\sigma\) (two standard deviations of the mean),
• Approximately 99.7% of the data lies within \( \mu\pm3\sigma\) (three standard deviations of the mean).

Here, \(\mu = 21000\) and \(\sigma=1000\).
\(\mu - 2\sigma=21000-2\times1000 = 19000\) and \(\mu=21000\)

Step2: Calculate the percentage

Since the normal distribution is symmetric about the mean \(\mu\). The interval from \(\mu - 2\sigma\) to \(\mu\) is half of the interval from \(\mu - 2\sigma\) to \(\mu + 2\sigma\).
The percentage of data within \(\mu\pm2\sigma\) is 95%. So the percentage of data within \(\mu - 2\sigma\) to \(\mu\) is \(\frac{95\%}{2}=47.5\%\)

Answer:

47.5