Applied Statistics and Probability for Engineers

8-7 TOLERABLE INTERVALS FOR A NORMAL DISTRIBUTION 271

One-sided tolerance bounds can also be computed. The tolerance factors for these bounds are

also given in Appendix Table XI.

EXAMPLE 8-9 Let’s reconsider the tensile adhesion tests originally described in Example 8-4. The load

at failure for n22 specimens was observed, and we found that 31.71 and s3.55.

We want to find a tolerance interval for the load at failure that includes 90% of the

values in the population with 95% confidence. From Appendix Table XI the tolerance

factorkfor n22, 0.90, and 95% confidence is k2.264. The desired tolerance

interval is

which reduces to (23.67, 39.75). We can be 95% confident that at least 90% of the values of

load at failure for this particular alloy lie between 23.67 and 39.75 megapascals.

From Appendix Table XI, we note that as , the value of kgoes to the z-value associated

with the desired level of containment for the normal distribution. For example, if we want

90% of the population to fall in the two-sided tolerance interval, kapproaches z0.051.645 as

. Note that as , a 100(1 )% prediction interval on a future value approaches a
tolerance interval that contains 100(1 )% of the distribution.

EXERCISES FOR SECTION 8-7

nS nS

nS

1 xks, xks 2 or 3 31.71 1 2.264 2 3.55, 31.71 1 2.264 2 3.55 4

x

8-60. Compute a 95% tolerance interval on the life of the

tires described in Exercise 8-22, that has confidence level

95%. Compare the length of the tolerance interval with the

length of the 95% CI on the population mean. Which interval

is shorter? Discuss the difference in interpretation of these

two intervals.

8-61. Consider the Izod impact test described in Exercise

8-23. Compute a 99% tolerance interval on the impact

strength of PVC pipe that has confidence level 90%.

Compare the length of the tolerance interval with the length

of the 99% CI on the population mean. Which interval is

shorter? Discuss the difference in interpretation of these two

intervals.

8-62. Compute a 99% tolerance interval on the brightness

of the television tubes in Exercise 8-24 that has confidence

level 95%. Compare the length of the prediction interval with

the length of the 99% CI on the population mean. Which

interval is shorter? Discuss the difference in interpretation of

these two intervals.

8-63. Consider the margarine test described in Exercise 8-25.

Compute a 99% tolerance interval on the polyunsaturated

fatty acid in this particular type of margarine that has confi-

dence level 95%. Compare the length of the prediction in-

terval with the length of the 99% CI on the population mean.

Which interval is shorter? Discuss the difference in inter-

pretation of these two intervals.

8-64. Compute a 90% tolerance interval on the compres-

sive strength of the concrete described in Exercise 8-26 that

has 90% confidence.

8-65. Compute a 95% tolerance interval on the diameter of

the rods described in Exercise 8-27 that has 90% confidence.

Compare the length of the prediction interval with the length

of the 95% CI on the population mean. Which interval is

shorter? Discuss the difference in interpretation of these two

intervals.

8-66. Consider the bottle wall thickness measurements

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

on bottle wall thickness that has confidence level 90%.

8-67. Consider the bottle wall thickness measurements

described in Exercise 8-29. Compute a 90% lower tolerance

bound on bottle wall thickness that has confidence level

90%. Why would a lower tolerance bound likely be of

interest here?

8-68. Consider the fuel rod enrichment data described in

Exercise 8-30. Compute a 99% tolerance interval on rod

enrichment that has confidence level 95%. Compare the

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

CI on the population mean.

8-69. Compute a 95% tolerance interval on the syrup vol-

ume described in Exercise 8-31 that has confidence level 90%.

Compare the length of the prediction interval with the length

of the 95% CI on the population mean.

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