Page 303 - Applied statistics and probability for engineers

Section 8-1/Conidence Interval on the Mean of a Normal Distribution, Variance Known 281

Exercises FOR SECTION 8-1

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion

8-1. For a normal population with known variance σ , (a) What is the value of the sample mean cycles to failure?
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answer the following questions: (b) The conidence level for one of these CIs is 95% and for
(a) What is the conidence level for the interval x − .2 14 σ n the other is 99%. Both CIs are calculated from the same
σ
≤ μ Ð x + .2 14 n ? sample data. Which is the 95% CI? Explain why.
(b) What is the conidence level for the interval x − . σ n 8-9. Suppose that n = 100 random samples of water from a
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9
2
x − . σ n ≤ μ ≤ x + . σ n ? freshwater lake were taken and the calcium concentration (mil-
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4
2
9
4
2
(c) What is the conidence level for the interval x − . σ1 85 n ligrams per liter) measured. A 95% CI on the mean calcium
≤ μ ≤ x + . σ n ? concentration is 0 49. ≤ μ ≤ 0 82.
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5
1
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(d) What is the conidence level for the interval μ ≤ x (a) Would a 99% CI calculated from the same sample data be
2 00σ n? longer or shorter?
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(e) What is the conidence level for the interval x −1 96 È n ≤μ? (b) Consider the following statement: There is a 95% chance
8-2. For a normal population with known variance σ : that μ is between 0.49 and 0.82. Is this statement correct?
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(a) What value of z /α 2 in Equation 8-5 gives 98% conidence? Explain your answer.
(b) What value of z /α 2 in Equation 8-5 gives 80% conidence? (c) Consider the following statement: If n = 100 random sam-
(c) What value of z /α 2 in Equation 8-5 gives 75% conidence? ples of water from the lake were taken and the 95% CI on
8-3. Consider the one-sided conidence interval expres- μ computed, and this process were repeated 1000 times,
sions for a mean of a normal population. 950 of the CIs would contain the true value of μ. Is this
(a) What value of z would result in a 90% CI? statement correct? Explain your answer.
α
(b) What value of z would result in a 95% CI? 8-10. Past experience has indicated that the breaking
α
(c) What value of z would result in a 99% CI? strength of yarn used in manufacturing drapery material is nor-
α
8-4. A conidence interval estimate is desired for the gain mally distributed and that σ = 2 psi. A random sample of nine
in a circuit on a semiconductor device. Assume that gain is nor- specimens is tested, and the average breaking strength is found
mally distributed with standard deviation s = 20. to be 98 psi. Find a 95% two-sided conidence interval on the
(a) Find a 95% CI for m when n = 10 and x = 1000 . true mean breaking strength.
(b) Find a 95% CI for m when n = 25 and x = 1000 . 8-11. The yield of a chemical process is being studied. From
(c) Find a 99% CI for m when n = 10 and x = 1000 . previous experience, yield is known to be normally distributed
(d) Find a 99% CI for m when n = 25 and x = 1000 . and σ = 3. The past ive days of plant operation have resulted in
the following percent yields: 91.6, 88.75, 90.8, 89.95, and 91.3.
(e) How does the length of the CIs computed change with the
Find a 95% two-sided conidence interval on the true mean yield.
changes in sample size and conidence level?
8-12. The diameter of holes for a cable harness is known to
8-5. A random sample has been taken from a normal distribu-
have a normal distribution with σ = 0.01 inch. A random sample
tion and the following conidence intervals constructed using the
of size 10 yields an average diameter of 1.5045 inch. Find a
same data: (38.02, 61.98) and (39.95, 60.05)
99% two-sided conidence interval on the mean hole diameter.
(a) What is the value of the sample mean?
8-13. A manufacturer produces piston rings for an auto-
(b) One of these intervals is a 95% CI and the other is a 90%
mobile engine. It is known that ring diameter is normally dis-
CI. Which one is the 95% CI and why?
tributed with σ = 0.001 millimeters. A random sample of 15
8-6. A random sample has been taken from a normal distribu- rings has a mean diameter of x = 74 .036 millimeters.
tion and the following conidence intervals constructed using the (a) Construct a 99% two-sided conidence interval on the mean
same data: (37.53, 49.87) and (35.59, 51.81) piston ring diameter.
(a) What is the value of the sample mean? (b) Construct a 99% lower-conidence bound on the mean pis-
(b) One of these intervals is a 99% CI and the other is a 95% ton ring diameter. Compare the lower bound of this coni-
CI. Which one is the 95% CI and why? dence interval with the one in part (a).
8-7. Consider the gain estimation problem in Exercise 8-4. 8-14. The life in hours of a 75-watt light bulb is known to be
(a) How large must n be if the length of the 95% CI is to be 40? normally distributed with σ = 25 hours. A random sample of 20
(b) How large must n be if the length of the 99% CI is to be 40?
bulbs has a mean life of x = 1014 hours.
8-8. Following are two conidence interval estimates of the (a) Construct a 95% two-sided conidence interval on the
mean m of the cycles to failure of an automotive door latch mean life.
mechanism (the test was conducted at an elevated stress level to (b) Construct a 95% lower-conidence bound on the mean life.
accelerate the failure). Compare the lower bound of this conidence interval with
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3124 9 ≤ μ ≤ 3215 7 3110 5 ≤ μ ≤ 3230 1 the one in part (a).
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