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264 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

It is also possible to find a 100(1 )% lower confidence bound or upper confidence bound
2
on .

2
The 100(1 )% lower and upper confidence bounds on are
2 2
1n 12s 2 2 1n 12s
and (8-22)
2 ,n 1 2 1 ,n 1

respectively.

EXAMPLE 8-5 An automatic filling machine is used to fill bottles with liquid detergent. A random sample of
2
2
20 bottles results in a sample variance of fill volume of s 0.0153 (fluid ounces) . If the
variance of fill volume is too large, an unacceptable proportion of bottles will be under- or
overfilled. We will assume that the fill volume is approximately normally distributed. A 95%
upper-confidence interval is found from Equation 8-22 as follows:

1n 12s 2
2

2
0.95,˛19
or

11920.0153
2
0.0287 1fluid ounce2 2
10.117
This last expression may be converted into a confidence interval on the standard deviation
by taking the square root of both sides, resulting in

0.17

Therefore, at the 95% level of confidence, the data indicate that the process standard deviation
could be as large as 0.17 fluid ounce.

EXERCISES FOR SECTION 8-4

8-33. Determine the values of the following percentiles: The sample standard deviation of the hole diameter measure-
2
2
2
2
2
2
2
0.05,10 , 0.025,15 , 0.01,12 , 0.95,20 , 0.99,18 , 0.995,16 , and 0.005,25 . ments is s 0.008 millimeters. Construct a 99% lower confi-
2
2
8-34. Determine the percentile that is required to dence bound for .
construct each of the following CIs: 8-36. The sugar content of the syrup in canned peaches is
(a) Confidence level 95%, degrees of freedom 24, normally distributed. A random sample of n 10 cans yields
one-sided (upper) a sample standard deviation of s 4.8 milligrams. Find a
(b) Confidence level 99%, degrees of freedom 9, one- 95% two-sided confidence interval for .
sided (lower) 8-37. Consider the tire life data in Exercise 8-22. Find a
(c) Confidence level 90%, degrees of freedom 19, two- 95% lower confidence bound for .
2
sided.
8-38. Consider the Izod impact test data in Exercise 8-23.
8-35. A rivet is to be inserted into a hole. A random sample Find a 99% two-sided confidence interval for .
2
of n 15 parts is selected, and the hole diameter is measured.