Fundamentals of Probability and Statistics for Engineers

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 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. D isplay these intervals graphically.

9.33 LetX 1 andX 2 be independent sample means from two normal populations
N(m 1 ,^21 )andN(m 2 ,^22 ), respectively. If^21 and^22 are known, show that a
[100(1 )]% confidence interval for m 1 m 2 is

where n 1 and n 2 are, respectively, the sample sizes from N(m 1 ,^22 )andN(m 2 ,^22 ),

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

9.34 Let us assume that random variable X in Problem 8.2(e) has a Poisson distribution
with pmf

Use the sample values of X given in Problem 8.2(e) (see Table 8.5, page 255)

and:

(a) D etermine M LE 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

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 1100 hours and the observed sample standard deviation s 50 hours.
(a) D etermine 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.

Parameter Estimation 313

ˆ

   

 

P…X 1 X 2 †u= 2

^21

n 1

‡

^22

n 2

 1 = 2

<m 1 m 2 <…X 1 X 2 †‡u= 2

^21

n 1

‡

^22

n 2

"# 1 = 2

ˆ 1 ;

 

PU>u /2)ˆ /2.

pX…k;†ˆ

ke

k!

; kˆ 0 ; 1 ; 2 ;...:

^ 



^



xˆ

1

30

X^30

iˆ 1

xiˆ 71 :3 years;

s^2 ˆ

1

29

X^30

iˆ 1

…xix†^2 ˆ 128 …years†^2 :



ˆ ˆ