(2) The highway miles per gallon rating of a certain model of truck is designed
ID: 3358948 • Letter: #
Question
(2) The highway miles per gallon rating of a certain model of truck is designed to be about 31 miles per gallon(mpg). The fuel efficiency that a driver obtains on an individual tank of gasoline naturally varies from tankful to tankful. Suppose the distribution of mpg calculations per tank is Normal and has a mean of 31 mpg and a standard deviation of 3 mpg. (a) What is the probability of getting a mean of 31.6 mpg or more on 16 random tanks of gas? (b) What should the sample size be so that at least 95% of the sample means are within 0.8 mpg of the population mean?
Explanation / Answer
the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
equation of the normal curve is ( Z )= x - u / sd/sqrt(n) ~ N(0,1)
mean of the sampling distribution ( x ) = 31
standard Deviation ( sd )= 3/ Sqrt ( 16 ) =0.75
sample size (n) = 16
a.
P(X < 31.6) = (31.6-31)/3/ Sqrt ( 16 )
= 0.6/0.75= 0.8
= P ( Z <0.8) From Standard NOrmal Table
= 0.78814
P(X > = 31.6) = 1 - P(X < 31.6)
= 1 - 0.78814 = 0.21186
b.
Compute Sample Size
n = (Z a/2 * S.D / ME ) ^2
Z/2 at 0.05% LOS is = 1.96 ( From Standard Normal Table )
Standard Deviation ( S.D) = 3
ME =0.8
n = ( 1.96*3/0.8) ^2
= (5.88/0.8 ) ^2
= 54.023 ~ 55
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