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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
x
at failure for n 22 specimens was observed, and we found that 31.71 and s 3.55.
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
factor k for n 22, 0.90, and 95% confidence is k 2.264. The desired tolerance
interval is

1x ks, x ks2 or 331.71 12.26423.55, 31.71 12.26423.554

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 n S
, the value of k goes 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, k approaches z 0.05 1.645 as
n S
. Note that as n S
, 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
8-60. Compute a 95% tolerance interval on the life of the 8-64. Compute a 90% tolerance interval on the compres-
tires described in Exercise 8-22, that has confidence level sive strength of the concrete described in Exercise 8-26 that
95%. Compare the length of the tolerance interval with the has 90% confidence.
length of the 95% CI on the population mean. Which interval 8-65. Compute a 95% tolerance interval on the diameter of
is shorter? Discuss the difference in interpretation of these the rods described in Exercise 8-27 that has 90% confidence.
two intervals. Compare the length of the prediction interval with the length
8-61. Consider the Izod impact test described in Exercise of the 95% CI on the population mean. Which interval is
8-23. Compute a 99% tolerance interval on the impact shorter? Discuss the difference in interpretation of these two
strength of PVC pipe that has confidence level 90%. intervals.
Compare the length of the tolerance interval with the length 8-66. Consider the bottle wall thickness measurements
of the 99% CI on the population mean. Which interval is described in Exercise 8-29. Compute a 90% tolerance interval
shorter? Discuss the difference in interpretation of these two on bottle wall thickness that has confidence level 90%.
intervals. 8-67. Consider the bottle wall thickness measurements
8-62. Compute a 99% tolerance interval on the brightness described in Exercise 8-29. Compute a 90% lower tolerance
of the television tubes in Exercise 8-24 that has confidence bound on bottle wall thickness that has confidence level
level 95%. Compare the length of the prediction interval with 90%. Why would a lower tolerance bound likely be of
the length of the 99% CI on the population mean. Which interest here?
interval is shorter? Discuss the difference in interpretation of 8-68. Consider the fuel rod enrichment data described in
these two intervals. Exercise 8-30. Compute a 99% tolerance interval on rod
8-63. Consider the margarine test described in Exercise 8-25. enrichment that has confidence level 95%. Compare the
Compute a 99% tolerance interval on the polyunsaturated length of the prediction interval with the length of the 95%
fatty acid in this particular type of margarine that has confi- CI on the population mean.
dence level 95%. Compare the length of the prediction in- 8-69. Compute a 95% tolerance interval on the syrup vol-
terval with the length of the 99% CI on the population mean. ume described in Exercise 8-31 that has confidence level 90%.
Which interval is shorter? Discuss the difference in inter- Compare the length of the prediction interval with the length
pretation of these two intervals. of the 95% CI on the population mean.