Question

Let X represent the amount of gas, in gallons, drivers put in their cars when fueling at the small downtown gas station. The model for this variable X is uniformly distributed between 4 and 12 gallons.

For this uniform distribution the mean amount of gas put in a car is 8 gallons and the standard deviation is about 2.3 gallons.

Your friend is looking at this model for the amount of gas and says: "I remember some rule from my statistics class last year ~ something about there being a 68% probability that the amount of gasoline a randomly selected driver will put in the car is within one standard deviation of the mean. So would that work in this case?"
Your short answer to your friend is 'No, it will not work in this case."

Formulate your more complete answer to your friend that includes both finding the actual probability that the amount of gasoline that a randomly selected driver will put in the car is within one standard deviation of the mean and an explanation as to why this is not consistent with the 68% value.

Answer 1

Solution

68 % is valid for normal distribution or bell shaped distribution

uniform distribution is not bell- shaped

hence this rule is not valid

X - unif(4,12)

P(8- 2.3 < X< 8 + 2.3)

= P(5.7 <X< 10.3)

= F(10.3) - F(5.7)

= (10.3 - 5)/ (12 - 4)

= 0.6625

P(X < x ) = F(x) = (x- a)/(b-a) when a<X< b here a = 4 , b = 12

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