Page 330 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 313

9.32 A total of 93 yearly Buffalo snowfall measurements are given in Problem 8.2(g)
(see Table 8.6, page 255). Assume that it is approximately normal with standard
deviation ˆ 26 inches. Determine 95% confidence intervals for the mean using
measurements of (a) 1909 to 1939, (b) 1909 to 1959, (c) 1909 to 1979, and (d) 1909
to 1999. Display these intervals graphically.
9.33 Let X 1 and X 2 be independent sample means from two normal populations
N(m 1 , ) and N(m 2 , ), respectively. If 2 1 2 2 2 1 and 2 2 are known, show that a
[100(1 )]% confidence interval for m 1 m 2 is

" 1=2 1=2 #
2 2 2 2

1 2 1 2
P X 1 X 2 † u =2 ‡ < m 1 m 2 < X 1 X 2 †‡ u =2 ‡ ˆ 1 ;
n 1 n 2 n 1 n 2
2
2

where n 1 and n 2 are, respectively, the sample sizes from N(m 1 , ) and N(m 2 , ),

2 2
and u /2 is the value of standardized normal random variable U such that

P U > u /2 ) ˆ /2.
9.34 Let us assume that random variable X in Problem 8.2(e) has a Poisson distribution
with pmf
k
e
p X k; †ˆ ; k ˆ 0; 1; 2; ... :
k!
Use the sample values of X given in Problem 8.2(e) (see Table 8.5, page 255)
and:
(a) Determine MLE ^ for .
(b) Determine a 95% confidence interval for using asymptotic properties of
MLE . ^
9.35 Assume that the lifespan of US males is normally distributed with unknown
mean m and unknown variance 2 . A sample of 30 mortality histories of US males
shows that
30
1 X
x ˆ x i ˆ 71:3 years;
30
iˆ1
30
1 X 2 2
2
s ˆ x i x† ˆ 128 years† :
29
iˆ1
Determine the observed values of 95% confidence intervals for m and 2 .
9.36 The life of light bulbs manufactured in a certain plant can be assumed to be
normally distributed. A sample of 15 light bulbs gives the observed sample mean
x ˆ 1100 hours and the observed sample standard deviation s ˆ 50 hours.
(a) Determine a 95% confidence interval for the average life.
(b) Determine two-sided and one-sided 95% confidence intervals for its
variance.
9.37 A total of 12 of 100 manufactured items examined are found to be defective.
(a) Find a 99% confidence interval for the proportion of defective items in the
manufacturing process.

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