Applied Statistics and Probability for Engineers

270 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

8-7 TOLERANCE INTERVALS FOR A NORMAL DISTRIBUTION

Consider a population of semiconductor processors. Suppose that the speed of these processors

has a normal distribution with mean 600 megahertz and standard deviation 30 mega-

hertz. Then the interval from 600 1.96(30) 541.2 to 600 1.96(30) 658.8 megahertz

captures the speed of 95% of the processors in this population because the interval from

1.96 to 1.96 captures 95% of the area under the standard normal curve. The interval from

z 2 to z 2 is called a tolerance interval.

If and are unknown, we can use the data from a random sample of size nto compute

and s, and then form the interval. However, because of sampling

variability in and s, it is likely that this interval will contain less than 95% of the values in

the population. The solution to this problem is to replace 1.96 by some value that will make

the proportion of the distribution contained in the interval 95% with some level of confidence.

Fortunately, it is easy to do this.

x

x 1 x1.96 s, x1.96 s 2

Atolerance intervalfor capturing at least % of the values in a normal distribution

with confidence level 100(1 )% is

where kis a tolerance interval factor found in Appendix Table XI. Values are given

for 90%, 95%, and 95% and for 95% and 99% confidence.

xks, xks

Definition

8-51. Consider the television tube brightness test described

in Exercise 8-24. Compute a 99% prediction interval on the

brightness of the next tube tested. Compare the length of the

prediction interval with the length of the 99% CI on the popu-

lation mean.

8-52. Consider the margarine test described in Exercise 8-25.

Compute a 99% prediction interval on the polyunsaturated

fatty acid in the next package of margarine that is tested.

Compare the length of the prediction interval with the length

of the 99% CI on the population mean.

8-53. Consider the test on the compressive strength of con-

crete described in Exercise 8-26. Compute a 90% prediction

interval on the next specimen of concrete tested.

8-54. Consider the suspension rod diameter measurements

described in Exercise 8-27. Compute a 95% prediction inter-

val on the diameter of the next rod tested. Compare the length

of the prediction interval with the length of the 95% CI on the

population mean.

8-55. Consider the bottle wall thickness measurements

described in Exercise 8-29. Compute a 90% prediction interval

on the wall thickness of the next bottle tested.

8-56. How would you obtain a one-sided prediction bound

on a future observation? Apply this procedure to obtain a 95%

one-sided prediction bound on the wall thickness of the next

bottle for the situation described in Exercise 8-29.

8-57. Consider the fuel rod enrichment data described

in Exercise 8-30. Compute a 99% prediction interval on the

enrichment of the next rod tested. Compare the length of the

prediction interval with the length of the 95% CI on the

population mean.

8-58. Consider the syrup dispensing measurements de-

scribed in Exercise 8-31. Compute a 95% prediction interval

on the syrup volume in the next beverage dispensed. Compare

the length of the prediction interval with the length of the 95%

CI on the population mean.

8-59. Consider the natural frequency of beams described

in Exercise 8-32. Compute a 90% prediction interval on the

diameter of the natural frequency of the next beam of this

type that will be tested. Compare the length of the prediction

interval with the length of the 95% CI on the population

mean.

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