Question

A major cab company in Chicago has computed its mean fare from O'Hare Airport to the Drake Hotel to be $25.89, with a standard deviation of $3.10. Based on this information, complete the following statements about the distribution of the company's fares from O'Hare Airport to the Drake Hotel.

(a) According to Chebyshev's theorem, at least ?56%75%84%89% of the fares lie between 19.69 dollars and 32.09 dollars.

(b) According to Chebyshev's theorem, at least 8/9 (about 89%) of the fares lie between dollars and dollars. (Round your answer to 2 decimal places.)

(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the fares lie between dollars and dollars.

(d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately ?68%75%95%99.7% of the fares lie between 19.69 dollars and 32.09 dollars.

Answer 1

We are given the mean here as 25.89 and standard deviation as 3.10

a) 19.69 = 25.89 - 2*3.1 that is 2 standard deviations less than the mean.
32.09 = 25.89 + 2*3.1 that is 2 standard deviation more than the mean.

By Chebyshev's inequality theorem, at least 1 - 1/k² that is 1 - 1/2² = 75% of the observations lies within 2 standard deviations of the mean.

Therefore 75% is the correct answer here.

b) 8/9 = 1 - 1/k²

1/k² = 1/9

k = 3

Therefore the limits here is computed as:
25.89 - 3*3.1 = 16.59
25.89 + 3*3.1 = 35.19

Therefore 16.59 and 35.19 are the limits here.

c) For bell shaped distribution, 68% of the observations lies within 1 standard deviation of the mean.

25.89 - 3.1 = 22.79
25.89 + 3.1 = 28.99

Therefore 22.79 and 28.99 are the limits here.

d) 25.89 + 2*3.1 = 32.09 and 19.69 = 25.89 - 2*3.1

For bell shaped distribution approx. 95% of the observations lies within 2 standard deviations of the mean. Therefore 95% is the required answer here.