Download HW Answers - TeacherWeb

HW Answers- p.603 #41, 42
41.
A bottling company uses a filling machine to fill plastic bottles with cola. The bottles are
supposed to contain 300 milliliters (ml). In fact, the contents vary according to a Normal
distribution with mean µ = 298 ml and standard deviation σ = 3 ml.
(a) What is the probability that an individual bottle contains less than 295 ml?
N (298,3)
295 − 298 

P ( X < 295 ) = P  z <
 = P ( z < −1) = .1587
3


(b) What is the probability that the mean contents of the bottles in a six-pack is less than
295 ml?
µ = 298
σ =3
Assumptions
n=6
x < 295
It is safe to assume the population of plastic
N ≥ 10n
cola bottles manufactured by a certain bottling
N ≥ 10(6)
company exceeds 60.
N ≥ 60
Normal Model
µx = µ
µ x = 298
σx =
σ
N (298,1.225)
n
3
σx =
= 1.225
6
Probability
295 − 298 

P ( x < 295 ) = P  z <
 = P ( z < −2.449 ) = .0072
1.225 

Interpret
The probability that, in a sample of six bottles, that the mean soda volume is less than
295ml is .0072.
42.
A company that owns and services a fleet of cars for its sales force has found that the service
lifetime of disc brake pads varies from car to car according to a Normal distribution with mean
µ = 55, 000 miles and standard deviation σ = 4500 miles. The company installs a new brand
of brake pads on 8 cars.
(a) If the new brand has the same lifetime distribution as the previous type (the one described
above), what is the distribution of the sample mean lifetime for the 8 cars?
4500 

N  55000,
 = N ( 55000,1590.99 )
8 

(b) The average life of the pads on these 8 cars turns out to be x = 51,800 miles. What is the
probability that the sample mean lifetime is 51,800 miles or less if the lifetime distribution is
unchanged? (The company takes this probability as evidence that the average lifetime of the
new brand of pads is less than 55,000 miles.)
µ = 55000
σ = 4500
Assumptions
N ≥ 10n
N ≥ 10(8)
It is safe to assume that the population of
the new brand of brake pads exceeds 80.
n=8
x < 51800
N ≥ 80
Normal Model
µx = µ
µ x = 55000
N ( 55000,1590.99 )
σx =
σ
n
4500
σx =
= 1590.99
8
Probability
51800 − 55000 

P ( x < 51800 ) = P  z <
 = P ( z < −2.011) = .0222
1590.99


Interpret
The probability that, in a sample of 8 cars, the mean lifetime of the new brake pads is less
than 51,800 miles is .0222. (Since this is so rare, only 2.22% of samples, this may lead the
company to believe that the manufacturer of the new brake pads are falsely advertising the
lifetime of the brake pads.)