Gasoline must contain a percentage X of a special additive. Specifications require that X is between 30 and 35 percent. The manufacturer will make a net profit (L), per gallon, which is given by the function of X: L (X) = $ 0.10 per gallon if 30 < X <35 L (X) = $ 0.05 per gallon if 35 ≤ X <40 or 25 < X ≤ 30 L (X) = - $ 0.10 per unit if x takes on any other value a) We know that X~N (33; 9). Calculate E (L). b) Suppose the manufacturer wants to increase its expected net profit by 50%. He intends to achieve this goal, increasing its net profit per gallon sold meeting the specifications (30 < X <35). What should the new net profit be?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Gasoline must contain a percentage X of a special additive. Specifications require that X is between 30 and 35 percent. The manufacturer will make a net profit (L), per gallon, which is given by the
L (X) = $ 0.10 per gallon if 30 < X <35
L (X) = $ 0.05 per gallon if 35 ≤ X <40 or 25 < X ≤ 30
L (X) = - $ 0.10 per unit if x takes on any other value
a) We know that X~N (33; 9). Calculate E (L).
b) Suppose the manufacturer wants to increase its expected net profit by 50%. He intends to achieve this goal, increasing its net profit per gallon sold meeting the specifications (30 < X <35). What should the new net profit be?
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