Page 323 - Applied Statistics And Probability For Engineers

c08.qxd 5/15/02 6:13 PM Page 275 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

8-7 TOLERABLE INTERVALS FOR A NORMAL DISTRIBUTION 275

MIND-EXPANDING EXERCISES

8-86. An electrical component has a time-to-failure and n is approximately
(or lifetime) distribution that is exponential with param-
2
eter , so the mean lifetime is 1 . Suppose that a 1 1 p ,4
sample of n of these components is put on test, and let n 2 a 1 p b a 4 b
X i be the observed lifetime of component i. The test con-
tinues only until the rth unit fails, where r n. This re-
sults in a censored life test. Let X 1 denote the time at (a) In order to be 95% confident that at least 90% of the
which the first failure occurred, X 2 denote the time at population will be included between the extreme
which the second failure occurred, and so on. Then the values of the sample, what sample size will be re-
total lifetime that has been accumulated at test termina- quired?
tion is (b) A random sample of 10 transistors gave the follow-
ing measurements on saturation current (in mil-
r
T r a X i 1n r2 X r liamps): 10.25, 10.41, 10.30, 10.26, 10.19, 10.37,
i 1 10.29, 10.34, 10.23, 10.38. Find the limits that con-
tain a proportion p of the saturation current meas-
We have previously shown in Exercise 7-72 that T r r is
urements at 95% confidence. What is the proportion
an unbiased estimator for .
p contained by these limits?
(a) It can be shown that 2 T r has a chi-square distribution
8-89. Suppose that X 1 , X 2 , p , X n is a random
with 2r degrees of freedom. Use this fact to develop a
sample from a continuous probability distribution
100(1 )% confidence interval for mean lifetime ~
with median .
1 .
(a) Show that
(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, P 5min 1X i 2 max 1X i 26
28, 29. Find a 95% confidence interval on mean 1 n 1
lifetime. 1 a b
2
8-87. Consider a two-sided confidence interval for
the mean when is known;
~
Hint: The complement of the event 3min 1X i 2
~
~
max 1X i 24 is 3max 1X i 2 4 ´ 3min 1X i 2 4, but
x z 1n x z 1n ~ ~
1 2 max1X i 2 if and only if X i for all i.4
(b) Write down a 100(1 )% confidence interval for
where 1 2 . If 1 2 2, we have the usual the median , where
~

100(1 )% confidence interval for . In the above,
, the interval is not symmetric about .
when 1 2 n 1
z 2 1n. 1
2
The length of the interval is L 1z 1 a b .
Prove that the length of the interval L is minimized when 2
1 2 2. Hint: Remember that 1z a 2 1 ,
1
so 11 2 z , and the relationship between the 8-90. Students in the industrial statistics lab at ASU
derivative of a function y f (x) and the inverse calculate a lot of confidence intervals on . Suppose all
1 1
x f 1 y2 is 1d dy2 f 1y2 1 31d dx2 f 1x24. these CIs are independent of each other. Consider the
8-88. It is possible to construct a nonparametric tol- next one thousand 95% confidence intervals that will be
erance interval that is based on the extreme values in a calculated. How many of these CIs do you expect to
random sample of size n from any continuous population. capture the true value of ? What is the probability that
If p is the minimum proportion of the population con- between 930 and 970 of these intervals contain the true
tained between the smallest and largest sample observa- value of ?
tions with confidence 1 , it can be shown that
n
np n 1 1n 12p