E1C04 09/14/2010
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4.9 For the data in each column, determine the sample mean value, standard deviation, and standard
deviation of the means. State the degrees of freedom in each.
4.10 Explain the concept of ‘‘central tendency’’ by comparing the range of the measured values and the
sample mean values from each of the three data sets.
4.11 From the data in column 1, estimate the range of values for which you would expect 95% of all
possible measured values for this operating condition to fall. Repeat for columns 2 and 3. Discuss
these outcomes in terms of what you might expect from finite statistics.
4.12 From the data in column 1, determine the best estimate of the mean value at a 95% probability level.
How does this estimate differ from the estimates made in problem 4.11? Repeat for columns 2 and 3.
Why do the estimates vary for each data set? Discuss these outcomes in terms of what you might expect
from finite statistics if these are measuring the same measured variable during the same process.
4.13 For the data in column 3, if one additional measurement were made, estimate the interval in which
the value of this measurement would fall with a 95% probability.
4.14 Compute a pooled sample mean value for the process. State the range for the best estimate in force at
95% probability based on these data sets. Discuss whether this pooled sample mean value is
reasonable given the sample mean values for the individual data sets. Write a short essay explanation
in terms of the limitations of sample statistics, the number of measurements, variations in data sets,
and statistical estimators.
4.15 Apply the x
2 goodness-of-fit test to the data in column 1 and test the assumption of a normal
distribution.
4.16 Consider a process in which the applied measured load has a known true mean of 100 N with variance
of 400 N
2 . An engineer takes 16 measurements at random. What is the probability that this sample
will have a mean value between 90 and 110?
4.17 A professor grades students on a normal curve. For any grade x, based on a course mean and standard
deviation developed over years of testing, the following applies:
A: x > x þ 1:6s
B: x þ 0:4s < x x þ 1:6s
C: x À 0:4s < x x þ 0:4s
D: x À 1:6s < x x À 0:4s
F: x x À 1:6s
How many A, C, and D grades are given per 100 students?
4.18 The production of a certain polymer fiber follows a normal distribution with a true mean diameter of
20 mm and a standard deviation of 30 mm. Compute the probability of a measured value greater than
80 mm. Compute the probability of a measured value between 50 and 80 mm.
4.19 An automotive manufacturer removes the friction linings from the clutch plates of drag race cars
following test runs. A sampling of 10 linings for wear show the following values (in mm): 204.5,
231.1, 157.5, 190.5, 261.6, 127.0, 216.6, 172.7, 243.8, and 291.0. Estimate the average wear and its
variance. Based on this sample, how many clutch plates out of a large set will be expected to show
wear of more than 203 mm?
4.20 Determine the mean value of the life of an electric light bulb if
p x
ð Þ ¼ 0:001e
À0:001x
x ! 0
and p(x) ¼ 0 otherwise. Here x is the life in hours.
Problems 155
14:7:49 Page 155
4.9 For the data in each column, determine the sample mean value, standard deviation, and standard
deviation of the means. State the degrees of freedom in each.
4.10 Explain the concept of ‘‘central tendency’’ by comparing the range of the measured values and the
sample mean values from each of the three data sets.
4.11 From the data in column 1, estimate the range of values for which you would expect 95% of all
possible measured values for this operating condition to fall. Repeat for columns 2 and 3. Discuss
these outcomes in terms of what you might expect from finite statistics.
4.12 From the data in column 1, determine the best estimate of the mean value at a 95% probability level.
How does this estimate differ from the estimates made in problem 4.11? Repeat for columns 2 and 3.
Why do the estimates vary for each data set? Discuss these outcomes in terms of what you might expect
from finite statistics if these are measuring the same measured variable during the same process.
4.13 For the data in column 3, if one additional measurement were made, estimate the interval in which
the value of this measurement would fall with a 95% probability.
4.14 Compute a pooled sample mean value for the process. State the range for the best estimate in force at
95% probability based on these data sets. Discuss whether this pooled sample mean value is
reasonable given the sample mean values for the individual data sets. Write a short essay explanation
in terms of the limitations of sample statistics, the number of measurements, variations in data sets,
and statistical estimators.
4.15 Apply the x
2 goodness-of-fit test to the data in column 1 and test the assumption of a normal
distribution.
4.16 Consider a process in which the applied measured load has a known true mean of 100 N with variance
of 400 N
2 . An engineer takes 16 measurements at random. What is the probability that this sample
will have a mean value between 90 and 110?
4.17 A professor grades students on a normal curve. For any grade x, based on a course mean and standard
deviation developed over years of testing, the following applies:
A: x > x þ 1:6s
B: x þ 0:4s < x x þ 1:6s
C: x À 0:4s < x x þ 0:4s
D: x À 1:6s < x x À 0:4s
F: x x À 1:6s
How many A, C, and D grades are given per 100 students?
4.18 The production of a certain polymer fiber follows a normal distribution with a true mean diameter of
20 mm and a standard deviation of 30 mm. Compute the probability of a measured value greater than
80 mm. Compute the probability of a measured value between 50 and 80 mm.
4.19 An automotive manufacturer removes the friction linings from the clutch plates of drag race cars
following test runs. A sampling of 10 linings for wear show the following values (in mm): 204.5,
231.1, 157.5, 190.5, 261.6, 127.0, 216.6, 172.7, 243.8, and 291.0. Estimate the average wear and its
variance. Based on this sample, how many clutch plates out of a large set will be expected to show
wear of more than 203 mm?
4.20 Determine the mean value of the life of an electric light bulb if
p x
ð Þ ¼ 0:001e
À0:001x
x ! 0
and p(x) ¼ 0 otherwise. Here x is the life in hours.
Problems 155