Question 1192350: The level of nitrogen oxides (NOX) in the exhaust of cars of a particular model varies Normally with mean 0.22 grams per mile (g/mi) and standard deviation 0.059 g/mi. Government regulations call for NOX emissions no higher than 0.28 g/mi.
A) What is the probability (±0.001) that a single-car of this model fails to meet the NOX requirement?
B) A company has 15 cars of this model in its fleet. What is the probability (±0.001) that the average NOX level x of these cars is above the 0.28 g/mi limit?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! mean = .22 grams per mile.
standard deviation = .059 grams per mile.
test level of NOX = 28 grams per mile.
z-score formula is z = (x - m) / s.
s represents the standard deviation when applied to a sample of one single element.
z-score formula is z = (x - m) / s.
s represents the standard error when applied to the mean of a sample of n elements.
the formula for standard error is that s = standard deviation divided by square root of sample size.
the first part of your problem is looking at a sample of one single element.
the z-score formula becomes:
z = (x - m) / s becomes z = (.28 - .22) / .059.
solve for z to get z = 1.016949153.
it's best to use a calculator to find the area to the right of that z-score.
using my ti-84 plus, i get the area to the right of that z-score = .1545888216.
that's the probability of getting a sample of one element having having NOX emissions greater than .28 grams per mile.
round that to 3 decimal places and it will be equal to .155.
the second part of your problem is looking at a sample of 15 elements.
here, you would use the standard error, rather than the standard deviation.
the standard error formula is s = standard deviation divided by the square root of the sample size.
that becomes s = .059 / sqrt(15) = .0152337345.
the z-score formula becomes z = (.28 - .22) / .0152337345 = 3.938627132.
using my ti-84 plus, i get the area to the right of that z-score = .0004099196854.
that's the probability of getting a sample of 15 elements having a mean of NOX emissions greater than .28 grams per mile.
round that to 3 decimal places and it will be equal to 0.
the mean of the sample will get closer to the mean of the population as the sample size gets larger.
that's a basic principle being applied through the use of the standard error formula.
since the mean of the population was .22, then it became extremely unlikely that the mean of the sample of 15 elements would be greater than .28.
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