Applied Statistics and Probability for Engineers

8-7 TOLERABLE INTERVALS FOR A NORMAL DISTRIBUTION 275

MIND-EXPANDING EXERCISES

8-86. An electrical component has a time-to-failure

(or lifetime) distribution that is exponential with param-

eter , so the mean lifetime is  1 . Suppose that a

sample of nof these components is put on test, and let

Xibe the observed lifetime of component i. The test con-

tinues only until the rth unit fails, where rn. This re-

sults in a censoredlife test. Let X 1 denote the time at

which the first failure occurred, X 2 denote the time at

which the second failure occurred, and so on. Then the

total lifetime that has been accumulated at test termina-

tion is

We have previously shown in Exercise 7-72 that Trris

an unbiased estimator for .

(a) It can be shown that 2Trhas a chi-square distribution

with 2rdegrees of freedom. Use this fact to develop a

100(1)% confidence interval for mean lifetime

 1 .

(b) Suppose 20 units were put on test, and the test

terminated after 10 failures occurred. The failure

times (in hours) are 15, 18, 19, 20, 21, 21, 22, 27,

28, 29. Find a 95% confidence interval on mean

lifetime.

8-87. Consider a two-sided confidence interval for

the mean when is known;

where  1  2  . If  1  2  2, we have the usual

100(1  )% confidence interval for . In the above,

when , the interval is not symmetric about .

The length of the interval is

Prove that the length of the interval Lis minimized when

 1  2 2. Hint:Remember that ,

so and the relationship between the

derivative of a function yf(x) and the inverse

is

8-88. It is possible to construct a nonparametric tol-

erance intervalthat is based on the extreme values in a

random sample of size nfrom any continuous population.

If pis the minimum proportion of the population con-

tained between the smallest and largest sample observa-

tions with confidence 1  , it can be shown that

and nis approximately

(a) In order to be 95% confident that at least 90% of the

population will be included between the extreme

values of the sample, what sample size will be re-

quired?

(b) A random sample of 10 transistors gave the follow-

ing measurements on saturation current (in mil-

liamps): 10.25, 10.41, 10.30, 10.26, 10.19, 10.37,

10.29, 10.34, 10.23, 10.38. Find the limits that con-

tain a proportion pof the saturation current meas-

urements at 95% confidence. What is the proportion

pcontained by these limits?

8-89. Suppose that X 1 , X 2 , p, Xnis a random

sample from a continuous probability distribution

with median

(a) Show that

Hint:The complement of the event

 is but

max if and only if for all i.

(b) Write down a 100(1  )% confidence interval for

the median , where

.

8-90. Students in the industrial statistics lab at ASU

calculate a lot of confidence intervals on . Suppose all

these CIs are independent of each other. Consider the

next one thousand 95% confidence intervals that will be

calculated. How many of these CIs do you expect to

capture the true value of ? What is the probability that

between 930 and 970 of these intervals contain the true

value of ?

a

1

2

b

n 1

~

1 Xi 2 ~ Xi~ 4

max 1 Xi 24 3 max 1 Xi 2 ~ 4 ́ 3 min 1 Xi 2 ~ 4 ,

3 min 1 Xi 2 ~

 1 a^12 b

n 1

P 5 min 1 Xi 2 ~max 1 Xi 26

~.

n^12 a

1 p

1 pb^ a

^2 ,4

4

b

npn^1  1 n 12 pn

xf ^11 y 2 1 d dy 2 f ^11 y 2  1 31 d dx 2 f 1 x 24.

^111  2 z,

 1 za 2  1 

L 1 z 1 z 22 1 n.

 1  2

xz 1  1 nxz 2  1 n

Tra

r

i 1

Xi 1 nr 2 Xr

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