Question

A major cab company in Chicago has computed its mean fare from O'Hare Airport to the Drake Hotel to be $29.82, with a standard deviation of $3.21. 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 ? ✓ of the fares lie between 25.005 dollars and 34.635 dollars. (b) According to Chebyshev's theorem, at least 2 v) of the fares lie between 23.40 dollars and 36.24 dollars. (c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 12V or the forestie between 23.40 dollars and 36.24 dollars (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the fares lie between dollars and dollars.

Answer 1

Chebyshev’s rule.  For any data set, the proportion (or percentage) of values that fall within  k  standard deviations from mean [ that is, in the interval  (
_{- ks, 7 + ks}
) ]  is at least   (
_{1 -2}
) , where  k > 1 .

(a)

From this we first have to find k,

Step 1 find the range = 34.635-25.005 = 9.63

Step 2 half the range to find central value = 4.815

Step 3 Divide Step 2 value by Standard deviation tp get the value of k

k = 4.815 / 3.21 = 1.5

Proportion or percentage is values lying between 25.005 and 34.635 are $1-\frac{1}{1.5^2}= \frac{5}{9} \approx 0.56$

(b)

From this we first have to find k,

Step 1 find the range = 36.24-23.40 = 12.84

Step 2 half the range to find central value = 6.42

Step 3 Divide Step 2 value by Standard deviation tp get the value of k

k = 6.42 / 3.21 = 2

Proportion or percentage is values lying between 25.005 and 34.635 are $1-\frac{1}{2^2}= \frac{3}{4} = 0.75$ = 75%

Empirical rule. For a data set with a symmetric distribution , approximately 68.3 percent of the values will fall within one standard deviation from the mean, approximately 95.4 percent will fall within 2 standard deviations from the mean, and approximately 99.7 percent will fall within 3 standard deviations from the mean.

(c) as we already know that the given data is two standard deviations away therefore 68.3 percent of data lie within this range.

(d) according to the empirical rule the range will be mean ± 1 standard deviation.

29.82 - 3.21 and 29.82+3.21

26.61 and 33.03

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