Question

A major cab company in Chicago has computed its mean fare from O'Hare Airport to the Drake Hotel to be $25.63, with a standard deviation of $4.05. Based on this information, complete the following statements about the distribution of the company's fares from O'Hare Airport to the Drake Hotel. of the fares lie (a) According to Chebyshev's theorem, at least ? between 17.53 dollars and 33.73 dollars. (b) According to Chebyshev's theorem, at least 84% of the fares lie between dollars and dollars. (Round your answer to 2 decimal places.) x 5 (C) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 99.7% of the fares lie between dollars and dollars. ? (d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximatel? the fares lie between 17.53 dollars and 33.73 dollars. 68% 75% 95% 99.7%

Answer 1

Let X be the random variable representing the fare from O'Hare Airport to the Drake hotel. Then the mean of X is given by $\mu=\$ 25.63$ nd standard deviation is given by $\sigma=\$ 4.05$ . Then the Chebychev's theorem can be stated as follows,

For any constant k, $P(\left | X-\mu \right |\leq k\sigma)\geq 1-\frac{1}{k^2}$ ..............................................(1)

(a) We want to find out the lower bound to the probability P(17.53<X<33.73).

$P(17.53<X<33.73)=P(17.53-\mu<X-\mu<33.73-\mu)=P(17.53-25.63<X-25.63<33.73-25.63)=P(-8.1<X-\mu<8.1)=P(\left | X-\mu \right |<8.1)$

comparing the final expression with (1) we get $k\sigma=8.1\Rightarrow k=\frac{8.1}{\sigma}=\frac{8.1}{4.05}=2$

Therefore $P(\left | X-25.63 \right |\leq 2\times 4.05)\geq 1-\frac{1}{2^2}=\frac{3}{4}=0.75$

According to Chebyshev's theorem , atleast 75% of the fares lie between 17.53 dollars and 33.73 dollars.

(b) Using the same procedures as above, we get $1-\frac{1}{k^2}=0.84\Rightarrow \frac{1}{k^2}=0.16\Rightarrow k^2=\frac{1}{0.16}\Rightarrow k=\frac{1}{0.4}=2.5$

Therefore substituting k=2.5 in (1) we get,$P(\left | X-25.63 \right |\leq 2.5\times 4.05)\geq 0.84$

$\Rightarrow P(25.63-( 2.5\times 4.05)\leq X\leq 25.63+( 2.5\times 4.05))\geq 0.84$

$\Rightarrow P(15.505\leq X\leq 35.755)\geq 0.84$

According to chebyshev's theorem atleast 84% of the fares lie between 15.505 dollars and 35.755 dollars.

(c)Suppose that the distribution is bell shaped 99.7% fares will lie between ($(\mu-3\sigma,\mu+3\sigma )$ limits.$(\mu-3\sigma,\mu+3\sigma ) =(25.63-3\times 4.05,25.63+3\times 4.05)=(13.48,37.78)$ limits

According to empirical rule ,approximately 99.7% of the fares lie between 13.48 dollars and 37.78 dollars.

(d) From part (a) we get $P(17.53<X<33.73)=P(\mu-2\sigma<X<\mu+2\sigma).$ So According to empirical rule $(\mu-2\sigma,\mu+2\sigma)$ would cover 95% of observations.

According to empirical rule approximately 95% of fairs lie between 17.53 dollars and 33.73 dollars.