the level of nitrogen oxides NOX in the exhaust of cars of a

the level of nitrogen oxides (NOX) in the exhaust of cars of a particular model varies Normally with mean 0.26 grams per mile (g/mi) and standard deviation 0.07 g/mi. Government regulations call for NOX emissions no higher than 0.32 g/mi.

(a) What is the probability that a single car of this model fails to meet the NOX requirement? (Use 3 decimal places)


(b) A company has 25 cars of this model in its fleet. What is the probability that the average NOX level of these cars is above the 0.32 g/mi limit? (Use 3 decimal places)


(c) What average NOX level for the fleet is less than 0.01 of all other average NOX levels? Hint: this requires a backward Normal calculation. (Use 2 decimal places)

Solution

(a) What is the probability that a single car of this model fails to meet the NOX requirement? (Use 3 decimal places)

P(X>0.3) = P((X-mean)/s >(0.3-0.24)/0.07)

=P(Z>0.86) =0.195 (from standard normal table)

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(b) A company has 25 cars of this model in its fleet. What is the probability that the average NOX level of these cars is above the 0.3 g/mi limit? (Use 3 decimal places)

P(xbar>0.3) = P((xbar-mean)/(s/vn) >(0.3-0.24)/(0.07/sqrt(25)))

=P(Z>4.29) =0 (from standard normal table)

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(c) What average NOX level for the fleet is less than 0.01 of all other average NOX levels? Hint: this requires a backward Normal calculation. (Use 2 decimal places)

P(X<c)=0.01

--> P(Z<(c-0.24)/0.07) = 0.01

--> (c-0.24)/0.07 = -2.33 (from standard normal table)

So c= 0.24-2.33*0.07 =0.08