Page 309 - Applied Statistics And Probability For Engineers

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8-4 CONFIDENCE INTERVAL ON THE VARIANCE AND STANDARD DEVIATION OF A NORMAL POPULATION 261

and standard deviation are 60,139.7 and 3645.94 kilometers. (a) Check the assumption of normality for rod diameter.
Find a 95% confidence interval on mean tire life. (b) Find a 95% two-sided confidence interval on mean rod
8-23. An Izod impact test was performed on 20 specimens of diameter.
x
PVC pipe. The sample mean is 1.25 and the sample stan- 8-28. Rework Exercise 8-27 to compute a 95% lower con-
dard deviation is s 0.25. Find a 99% lower confidence fidence bound on rod diameter. Compare this bound with the
bound on Izod impact strength. lower limit of the two-sided confidence limit from Exercise
8-24. The brightness of a television picture tube can be eval- 8-27. Discuss why they are different.
uated by measuring the amount of current required to achieve a 8-29. The wall thickness of 25 glass 2-liter bottles was meas-
particular brightness level. A sample of 10 tubes results in ured by a quality-control engineer. The sample mean was
x 317.2 and s 15.7. Find (in microamps) a 99% confi- x 4.05 millimeters, and the sample standard deviation was
dence interval on mean current required. State any necessary s 0.08 millimeter. Find a 95% lower confidence bound for
assumptions about the underlying distribution of the data. mean wall thickness. Interpret the interval you have obtained.
8-25. A particular brand of diet margarine was analyzed to 8-30. An article in Nuclear Engineering International
determine the level of polyunsaturated fatty acid (in percent- (February 1988, p. 33) describes several characteristics of fuel
ages). A sample of six packages resulted in the following data: rods used in a reactor owned by an electric utility in Norway.
16.8, 17.2, 17.4, 16.9, 16.5, 17.1. Measurements on the percentage of enrichment of 12 rods
(a) Is there evidence to support the assumption that the level were reported as follows:
of polyunsaturated fatty acid is normally distributed?
(b) Find a 99% confidence interval on the mean . Provide a 2.94 3.00 2.90 2.75 3.00 2.95
practical interpretation of this interval. 2.90 2.75 2.95 2.82 2.81 3.05
8-26. The compressive strength of concrete is being tested
(a) Use a normal probability plot to check the normality as-
by a civil engineer. He tests 12 specimens and obtains the
sumption.
following data.
(b) Find a 99% two-sided confidence interval on the mean
2216 2237 2249 2204 percentage of enrichment. Are you comfortable with the
statement that the mean percentage of enrichment is 2.95
2225 2301 2281 2263
percent? Why?
2318 2255 2275 2295
8-31. A postmix beverage machine is adjusted to release a
(a) Is there evidence to support the assumption that compres- certain amount of syrup into a chamber where it is mixed with
sive strength is normally distributed? Does this data set carbonated water. A random sample of 25 beverages was
support your point of view? Include a graphical display in found to have a mean syrup content of x 1.10 fluid ounces
your answer. and a standard deviation of s 0.015 fluid ounces. Find a
(b) Construct a 95% two-sided confidence interval on the 95% CI on the mean volume of syrup dispensed.
mean strength. 8-32. An article in the Journal of Composite Materials
(c) Construct a 95% lower-confidence bound on the mean (December 1989, Vol 23, p. 1200) describes the effect of delam-
strength. ination on the natural frequency of beams made from composite
8-27. A machine produces metal rods used in an automobile laminates. Five such delaminated beams were subjected to loads,
suspension system. A random sample of 15 rods is selected, and the resulting frequencies were as follows (in hertz):
and the diameter is measured. The resulting data (in millime-
230.66, 233.05, 232.58, 229.48, 232.58
ters) are as follows:
Find a 90% two-sided confidence interval on mean natural
8.24 8.25 8.20 8.23 8.24 frequency. Is there evidence to support the assumption of nor-
8.21 8.26 8.26 8.20 8.25 mality in the population?
8.23 8.23 8.19 8.28 8.24

8-4 CONFIDENCE INTERVAL ON THE VARIANCE AND
STANDARD DEVIATION OF A NORMAL POPULATION

Sometimes confidence intervals on the population variance or standard deviation are needed.
When the population is modeled by a normal distribution, the tests and intervals described in
this section are applicable. The following result provides the basis of constructing these con-
fidence intervals.