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 $16,000 and the standard deviation is $500. use the 68 - 95 - 99.7 rule to find the percentage of buyers who paid between $14,500 and $16,000. what percentage of buyers paid between $14,500 and $16,000? %

the figure illustrates a normal distribution for the prices paid for a particular model of a new car. the mean is $16,000 and the standard deviation is $500. use the 68 - 95 - 99.7 rule to find the percentage of buyers who paid between $14,500 and $16,000. what percentage of buyers paid between $14,500 and $16,000? %

Answer

Explanation:

Step1: Calculate the number of standard deviations from the mean

The mean is ( \mu = 16000 ) and the standard deviation is ( \sigma=500 ). For ( x = 14500 ), the number of standard deviations ( z=\frac{\mu - x}{\sigma}=\frac{16000 - 14500}{500}=\frac{1500}{500}=3 )

Step2: Apply the 68 - 95 - 99.7 Rule

The 68 - 95 - 99.7 Rule states that:

  • Approximately 68% of the data lies within ( \mu\pm\sigma )
  • Approximately 95% of the data lies within ( \mu\pm2\sigma )
  • Approximately 99.7% of the data lies within ( \mu\pm3\sigma )

The interval from ( \mu - 3\sigma) to ( \mu) is half of the interval from ( \mu - 3\sigma) to ( \mu+3\sigma).

Since the normal distribution is symmetric about the mean, the percentage of data from ( \mu - 3\sigma) to ( \mu) is ( \frac{99.7%}{2}=49.85%)

Answer:

(49.85)