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production. Production rates are 1000 per minute. What action do you recommend if such
a trial sampling of 30 bearings shows a mean of 5.060 mm with a standard deviation of 0.0025 mm?
4.36 Referring to the information in Problem 4.35, suppose the mean diameter of the bearings must be
held to within Æ0.2%. Based on given information, how many bearings should be measured at each
10-minute interval?
4.37 A manufacturer of general aircraft dry vacuum pumps wishes to estimate the mean failure time of its
product at 95% confidence. Initially, six pumps are tested to failure with these results (in hours of
operation): 1272, 1384, 1543, 1465, 1250, and 1319. Estimate the sample mean and the 95%
confidence interval of the true mean. How many more data points would be needed to improve the
confidence interval to be within Æ50 hours?
4.38 Strength tests on a batch of cold-drawn steel yield the following:
Strength (MPa)
Occurrences
421–480
4
481–540
8
541–600
12
601–660
6
Test the hypothesis that the data are a random sample that follows a normal distribution.
4.39 Determine the one-sided 95th-percentile of the chi-squared distribution at 16 degrees of freedom.
Refer to Figure 4.8.
4.40 The manufacturer of an electrical fuse claims that the fuse’s link will break with a 10% electrical
overload within 12.31 minutes. Independent tests on a random sampling of 20 fuses showed the mean
time for link breakage was 10.59 minutes with a standard deviation of 2.61 minutes. Does the test
support or refute the manufacturer’s claim?
4.41 How many independent random measurements of a time-dependent acceleration signal obtained
from a vibrating vehicle under fixed operating conditions would lead to a confidence interval about
the mean of Æ0.5 g, if the standard deviation of the signal is expected to be 1 g?
4.42 Based on 51 measurements of a time-dependent electrical signal, the standard deviation is 1.52 V.
Find the 95% confidence interval in its true mean value. How many more measurements would be
required to provide a 95% confidence interval in the true mean to within Æ0.28 V?
4.43 Estimate the probability with which an observed x
2
¼ 19.0 will be exceeded if there are 10 degrees of
freedom.
4.44 A company produces a computer battery that it claims last for 8 hours on a single charge with a
standard deviation of 20 minutes. A random sampling of seven batteries are tested and found to have
a sample standard deviation of 29 minutes. What is the chi-squared value and the predicted level of
significance as represented by the test?
4.45 A conductor is insulated using an enameling process. It is known that the number of insulation breaks
per meter of wire is 0.07. What is the probability of finding x breaks in a piece of wire 5 m long? Use
the Poisson distribution.
4.46 We know that 2% of the screws made by a certain machine are defective with the defects occurring
randomly. The screws are packaged 100 to a box. Estimate the probability that a box will contain x
defective screws. (a) Use the Binomial distribution. (b) Use the Poisson distribution.
158 Chapter 4 Probability and Statistics
14:7:49 Page 158
production. Production rates are 1000 per minute. What action do you recommend if such
a trial sampling of 30 bearings shows a mean of 5.060 mm with a standard deviation of 0.0025 mm?
4.36 Referring to the information in Problem 4.35, suppose the mean diameter of the bearings must be
held to within Æ0.2%. Based on given information, how many bearings should be measured at each
10-minute interval?
4.37 A manufacturer of general aircraft dry vacuum pumps wishes to estimate the mean failure time of its
product at 95% confidence. Initially, six pumps are tested to failure with these results (in hours of
operation): 1272, 1384, 1543, 1465, 1250, and 1319. Estimate the sample mean and the 95%
confidence interval of the true mean. How many more data points would be needed to improve the
confidence interval to be within Æ50 hours?
4.38 Strength tests on a batch of cold-drawn steel yield the following:
Strength (MPa)
Occurrences
421–480
4
481–540
8
541–600
12
601–660
6
Test the hypothesis that the data are a random sample that follows a normal distribution.
4.39 Determine the one-sided 95th-percentile of the chi-squared distribution at 16 degrees of freedom.
Refer to Figure 4.8.
4.40 The manufacturer of an electrical fuse claims that the fuse’s link will break with a 10% electrical
overload within 12.31 minutes. Independent tests on a random sampling of 20 fuses showed the mean
time for link breakage was 10.59 minutes with a standard deviation of 2.61 minutes. Does the test
support or refute the manufacturer’s claim?
4.41 How many independent random measurements of a time-dependent acceleration signal obtained
from a vibrating vehicle under fixed operating conditions would lead to a confidence interval about
the mean of Æ0.5 g, if the standard deviation of the signal is expected to be 1 g?
4.42 Based on 51 measurements of a time-dependent electrical signal, the standard deviation is 1.52 V.
Find the 95% confidence interval in its true mean value. How many more measurements would be
required to provide a 95% confidence interval in the true mean to within Æ0.28 V?
4.43 Estimate the probability with which an observed x
2
¼ 19.0 will be exceeded if there are 10 degrees of
freedom.
4.44 A company produces a computer battery that it claims last for 8 hours on a single charge with a
standard deviation of 20 minutes. A random sampling of seven batteries are tested and found to have
a sample standard deviation of 29 minutes. What is the chi-squared value and the predicted level of
significance as represented by the test?
4.45 A conductor is insulated using an enameling process. It is known that the number of insulation breaks
per meter of wire is 0.07. What is the probability of finding x breaks in a piece of wire 5 m long? Use
the Poisson distribution.
4.46 We know that 2% of the screws made by a certain machine are defective with the defects occurring
randomly. The screws are packaged 100 to a box. Estimate the probability that a box will contain x
defective screws. (a) Use the Binomial distribution. (b) Use the Poisson distribution.
158 Chapter 4 Probability and Statistics