In 2017, the entire fleet of light‑duty vehicles sold in the United States by each manufacturer must emit an average of no more than 9090 milligrams per mile (mg/mi) of nitrogen oxides (NOX) and nonmethane organic gas (NMOG) over the useful life (150,000150,000 miles of driving) of the vehicle. NOX ++ NMOG emissions over the useful life for one car model vary Normally with mean 8484 mg/mi and standard deviation 66 mg/mi. (a) What is the probability that a single car of this model emits more than 9090 mg/mi of NOX ++ NMOG? (Enter your answer rounded to four decimal places.) (b) A company has 1616 cars of this model in its fleet. What is the probability that the average NOX ++ NMOG level ?¯x¯ of these cars is above 9090 mg/mi? (Enter your answer rounded to four decimal places.)
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!
In 2017, the entire fleet of light‑duty vehicles sold in the United States by each manufacturer must emit an average of no more than 9090 milligrams per mile (mg/mi) of nitrogen oxides (NOX) and nonmethane organic gas (NMOG) over the useful life (150,000150,000 miles of driving) of the vehicle. NOX ++ NMOG emissions over the useful life for one car model vary Normally with mean 8484 mg/mi and standard deviation 66 mg/mi.
(a) What is the probability that a single car of this model emits more than 9090 mg/mi of NOX ++ NMOG? (Enter your answer rounded to four decimal places.)
(b) A company has 1616 cars of this model in its fleet. What is the probability that the average NOX ++ NMOG level ?¯x¯ of these cars is above 9090 mg/mi? (Enter your answer rounded to four decimal places.)
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