Question

A car salesman tells you that the model you are looking at gets on average 36 miles per gallon overall (highway and street driving) with a standard deviation of 4 miles per gallon. He also tells you that 20% of the cars get at
least 44 miles per gallon.

a. ( Do you think this is a normal distribution? Explain why or why not. (there is no one right answer for this)

b. Analyze the salesman's statement mathematically. Is he being at all untruthful?

i. Explain under the assumption that the distribution IS a normal distribution.

ii. Explain under the assumption that the distribution is NOT a normal distribution.

Answer 1

a) 44 can be written here as:
44 = 36 + 2*4 that is 2 standard deviations more than the mean.

Therefore 20% of the observations lying above 2 standard deviations above mean does not seem to be correct here ( especially if the distribution is normal). Still it can be possible here for some kind of skewed distributions.

b) (i) For a normal distribution, we have from the standard normal tables,
P(Z > 2) = 0.0228 which is way lower than 0.2, therefore the salesman's statement mathematically is not accepted here for a normal distribution. Therefore if the distribution is normal, then mathematically the salesman is being untruthful here.

(ii) For a non normal distribution, using chebyshev's inequality theorem, we know that at least 1 - 1/k² of the observations lies within k standard deviations of the mean.

Therefore, for k = 2, we have here: 1 - 1/k² = 1 - 1/2² = 0.75 that is 75% of the observations lies within 2 standard deviations of the mean. Therefore max (1 - 0.75)/2 = 12.5% of the observations lies above 44, but we are given here as 20% lying above 44. Therefore salesman is untruthful even when the underlying distribution is not normal.