Question

At one point the average price of regular unleaded gasoline was

​$3.543.54

per gallon. Assume that the standard deviation price per gallon is

​$0.080.08

per gallon and use​ Chebyshev's inequality to answer the following.

​(a) What percentage of gasoline stations had prices within

33

standard deviations of the​ mean?

​(b) What percentage of gasoline stations had prices within

2.52.5

standard deviations of the​ mean? What are the gasoline prices that are within

2.52.5

standard deviations of the​ mean?

​(c) What is the minimum percentage of gasoline stations that had prices between

​$3.383.38

and

​$3.703.70​?

Answer 1

Solution :

Given that mean x-bar = 3.54 , standard deviation s = 0.08

=> Use chebyshev's inequality
  
1) x-bar - k*s < values < x-bar + k*s

2) Proportion : (1 - 1/k^2)*100%

where x-bar = mean , k = # of standard deviations , s = standard deviation

(a) within 3 standard deviations of the​ mean

=> k = 3

=> (1 - 1/k^2)*100%

=> (1 - 1/3^2)*100%

=> 88.89%

(b) within 2.5 standard deviations of the​ mean and range

=> k = 2.5

=> (1 - 1/k^2)*100%

=> (1 - 1/2.5^2)*100%

=> 84%

=> Range is x-bar - k*s to x-bar + k*s

=> 3.54 - 2.5*0.08 to 3.54 + 2.5*0.08

=> 3.34 to 3.74

(c) between $3.38 and $3.70

=> x-bar - k*s = 3.38 and x-bar + k*s = 3.70

=> 3.54 - k*0.08 = 3.38 and 3.54 + k*0.08 = 3.70

=> k*0.08 = 3.54 - 3.38 and k*0.08 = 3.54 - 3.70

=> k = 0.16/0.08 and k = -0.16/0.08

=> 2 and k = -2

=> proportion : (1 - 1/k^2)*100%

=> (1 - 1/2^2)*100%

=> at least 75%