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Answer :
We would expect the central 95% of autos to be found in the interval from 20 to 40 mpg.
The 68-95-99.7 rule, also known as the empirical rule, states that for a normal distribution, approximately 68% of the observations fall within one standard deviation of the mean, approximately 95% of the observations fall within two standard deviations of the mean, and approximately 99.7% of the observations fall within three standard deviations of the mean.
Assuming that the distribution of autos' miles per gallon (mpg) is approximately normal, we can use the empirical rule to estimate the interval in which we would expect the central 95% of autos to be found.
Then, according to the empirical rule, approximately 95% of autos should be found within 2 standard deviations of the mean. We can write:
μ - 2σ ≤ x ≤ μ + 2σ
where x represents the mpg of an individual auto.
For example, if we take a random sample of 100 autos and find that the mean mpg is 30 and the standard deviation is 5, then we can estimate the interval in which we would expect the central 95% of autos to be found as:
30 - 2(5) ≤ x ≤ 30 + 2(5)
20 ≤ x ≤ 40
Therefore, we would expect the central 95% of autos to be found in the interval from 20 to 40 mpg.
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Final answer:
The central 95% of autos can be expected to be found within two standard deviations of the mean.
Explanation:
The central 95% of autos can be expected to be found within two standard deviations of the mean. Using the empirical rule, this means that the central 95% of autos would be expected to be found between the mean minus two standard deviations and the mean plus two standard deviations.
In this case, the mean fuel economy is 34.6 mpg and the standard deviation is 10.3 mpg. So, the interval in which the central 95% of autos would be expected to be found is:
34.6 - (2 * 10.3) to 34.6 + (2 * 10.3)
Therefore, the interval would be from 13.6 mpg to 55.6 mpg.
