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256 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

for , and (3) has standard deviation ˆ that can be estimated from the sample data, then the
ˆ
quantity 1 2 ˆ has an approximate standard normal distribution. Then a large-sample
approximate CI for is given by

ˆ
ˆ
z 2 ˆ z 2 ˆ (8-14)

Maximum likelihood estimators usually satisfy the three conditions listed above, so Equation
ˆ
8-14 is often used when is the maximum likelihood estimator of . Finally, note that
Equation 8-14 can be used even when ˆ is a function of other unknown parameters (or of ).
Essentially, all one does is to use the sample data to compute estimates of the unknown
parameters and substitute those estimates into the expression for ˆ .

8-2.6 Bootstrap Confidence Intervals (CD Only)

EXERCISES FOR SECTION 8-2
2
8-1. For a normal population with known variance , (a) What is the value of the sample mean cycles to failure?
answer the following questions: (b) The confidence level for one of these CIs is 95% and the
(a) What is the confidence level for the interval x 2.14 1n confidence level for the other is 99%. Both CIs are calcu-
x 2.14 1n ? lated from the same sample data. Which is the 95% CI?
(b) What is the confidence level for the interval x 2.49 1n Explain why.
x 2.49 1n ? 8-7. n 100 random samples of water from a fresh water
(c) What is the confidence level for the interval x 1.85 1n lake were taken and the calcium concentration (milligrams
x 1.85 1n ? per liter) measured. A 95% CI on the mean calcium concen-
2
8-2. For a normal population with known variance : tration is 0.49 0.82.
(a) What value of z 2 in Equation 8-7 gives 98% confidence? (a) Would a 99% CI calculated from the same sample data
(b) What value of z 2 in Equation 8-7 gives 80% confidence? been longer or shorter?
(c) What value of z 2 in Equation 8-7 gives 75% confidence? (b) Consider the following statement: There is a 95% chance
8-3. Consider the one-sided confidence interval expres- that is between 0.49 and 0.82. Is this statement correct?
sions, Equations 8-9 and 8-10. Explain your answer.
(a) What value of z would result in a 90% CI? (c) Consider the following statement: If n 100 random
(b) What value of z would result in a 95% CI? samples of water from the lake were taken and the 95% CI
(c) What value of z would result in a 99% CI? on computed, and this process was repeated 1000 times,
950 of the CIs will contain the true value of . Is this state-
8-4. A confidence interval estimate is desired for the gain in
ment correct? Explain your answer.
a circuit on a semiconductor device. Assume that gain is nor-
8-8. The breaking strength of yarn used in manufacturing
mally distributed with standard deviation 20.
(a) Find a 95% CI for when n 10 and x 1000. drapery material is required to be at least 100 psi. Past experi-
(b) Find a 95% CI for when n 25 and x 1000. ence has indicated that breaking strength is normally distrib-
(c) Find a 99% CI for when n 10 and x 1000. uted and that 2 psi. A random sample of nine specimens
(d) Find a 99% CI for when n 25 and x 1000. is tested, and the average breaking strength is found to be 98
psi. Find a 95% two-sided confidence interval on the true
8-5. Consider the gain estimation problem in Exercise 8-4.
mean breaking strength.
How large must n be if the length of the 95% CI is to be 40?
8-9. The yield of a chemical process is being studied. From
8-6. Following are two confidence interval estimates of the
previous experience yield is known to be normally distributed
mean of the cycles to failure of an automotive door latch
and 3. The past five days of plant operation have resulted
mechanism (the test was conducted at an elevated stress level
in the following percent yields: 91.6, 88.75, 90.8, 89.95, and
to accelerate the failure).
91.3. Find a 95% two-sided confidence interval on the true
mean yield.
3124.9 3215.7 3110.5 3230.1