The Environmental Protection Agency requires that the exhaust of each model of motor vehicle be tested for the level of several pollutants. The level of oxides of nitrogen (NOX) in the exhaust of one light truck model was found to vary among individual trucks according to a Normal distribution with mean 1.45 grams per mile driven and standard deviation 0.40 grams per mile.
(a) What is the 45th percentile for NOX exhaust, rounded to four decimal places?
(b) Find the interquartile range for the distribution of NOX levels in the exhaust of trucks rounded to four decimal places.
Solution:-
Given that,
mean = = 1.45 grams
standard deviation = = 0.40 grams
a) Using standard normal table,
P(Z < z) = 45%
= P(Z < z ) = 0.45
= P(Z < -0.1257 ) = 0.45
z = -0.1257
Using z-score formula,
x = z * +
x = -0.1257 * 0.40 + 1.45
x = 1.3997 grams.
b) Using standard normal table,
The z dist'n First quartile is,
P(Z < z) = 25%
= P(Z < z) = 0.25
= P(Z < -0.6745 ) = 0.25
z = -0.6745
Using z-score formula,
x = z * +
x = -0.6745 * 0.40 + 1.45
x = 1.1802 grams.
First quartile =Q1 = 1.1802 grams
The z dist'n Third quartile is,
P(Z < z) = 75%
= P(Z < z) = 0.75
= P(Z < 0.6745 ) = 0.75
z = 0.6745
Using z-score formula,
x = z * +
x = 0.6745 * 0.40 + 1.45
x = 1.7198 grams
Third quartile =Q3 = 1.7198 grams
IQR = Q3 - Q1
IQR = 1.7198 - 1.1802
IQR = 0.5396 grams.
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