Answer:
a) 95%
b) $16,000 to $28,000
c) 84%
d) 2.5%
Explanation:
Given the mean of car costs is $22,000 with a standard deviation of $2,000, you want to use the empirical rule to find ...
- percentage of buyers paying $18–26 thousand
- range of values for middle 99.7% of costs
- probability of cost less than $24,000
- probability of cost more than $26,000
Empirical rule
The empirical rule tells you that the center 68% of costs will be between -1 and +1 standard deviations from the mean: $20,000 to $24,000.
95% of costs will lie within 2 standard deviations: $18,000 to $26,000.
The "tails" of the distribution are split equally between the upper values of these ranges and the lower values.
a) 18-26
These values are ±2 standard deviations from the mean.
95% of buyers will pay between $18 and 26 thousand.
b) 99.7%
The middle 99.7% of the distribution lies between ±3 standard deviations from the mean:
22,000 ± 3(2000) = 22,000 ± 6,000 = {16000, 28000}
The middle 99.7% of costs are between $16,000 and $28,000.
c) < 24
We know that 68% of costs are between $20,000 and $24,000, and 50% of costs are below $22,000. The distribution is symmetrical, so 68%/2 = 34% of costs are between $22,000 and $24,000..
The fraction below $24,000 is ...
P(<24) = P(<22) +P(22 to 24) = 0.5 + 0.34 = 0.84
The probability a car will cost less than $24,000 is about 84%.
d) > 26
The empirical rule tells us 95% of the distribution is between 18 and 26 thousand. Half the remaining amount is above 26 thousand.
P(> 26) = (1 -0.95)/2 = 2.5%
The probability a car will cost more than $26,000 is about 2.5%.
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