Solved on Nov 18, 2023

Using the normal distribution, find the percentage of buyers who paid between $14,000\$14,000 and $18,000\$18,000 for a car with mean $18,000\$18,000 and standard deviation $2,000\$2,000.

STEP 1

Assumptions1. The distribution of car prices is normal. The mean price is $18,000\$18,000
3. The standard deviation is $,000\$,000
4. We are using the68-95-99.7 Rule (also known as the empirical rule), which states that for a normal distribution, approximately68% of the data falls within one standard deviation of the mean,95% falls within two standard deviations, and99.7% falls within three standard deviations.

STEP 2

We need to find the number of standard deviations away from the mean the values $14,000\$14,000 and $18,000\$18,000 are. We can do this by subtracting the mean from each value and dividing by the standard deviation.
Z=XμσZ = \frac{X - \mu}{\sigma}where ZZ is the number of standard deviations, XX is the value, μ\mu is the mean, and σ\sigma is the standard deviation.

STEP 3

Calculate the number of standard deviations for $14,000\$14,000.
Z14000=$14,000$18,000$2,000Z_{14000} = \frac{\$14,000 - \$18,000}{\$2,000}

STEP 4

Calculate the result.
Z14000=$14,000$18,000$2,000=2Z_{14000} = \frac{\$14,000 - \$18,000}{\$2,000} = -2This means that $14,000\$14,000 is2 standard deviations below the mean.

STEP 5

Calculate the number of standard deviations for $18,000\$18,000.
Z18000=$18,000$18,000$2,000Z_{18000} = \frac{\$18,000 - \$18,000}{\$2,000}

STEP 6

Calculate the result.
Z18000=$18,000$18,000$2,000=0Z_{18000} = \frac{\$18,000 - \$18,000}{\$2,000} =0This means that $18,000\$18,000 is0 standard deviations away from the mean, i.e., it is the mean.

STEP 7

Now, we need to use the68-95-99.7 Rule to find the percentage of buyers who paid between $14,000\$14,000 and $18,000\$18,000. According to the rule,95% of the data falls within two standard deviations of the mean. Since $14,000\$14,000 is2 standard deviations below the mean and $18,000\$18,000 is the mean, we are looking at half of this95% range.

STEP 8

Calculate the percentage of buyers who paid between $14,000\$14,000 and $18,000\$18,000.
Percentage=95%2Percentage = \frac{95\%}{2}

STEP 9

Calculate the result.
Percentage=95%2=47.5%Percentage = \frac{95\%}{2} =47.5\%The percentage of buyers who paid between $14,000\$14,000 and $18,000\$18,000 is47.5%.

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