Page 321 - Applied statistics and probability for engineers

Section 8-7/Tolerance and Prediction Intervals 299

Example 8-12 Alloy Adhesion Let’s reconsider the tensile adhesion tests originally described in Example 8-6.
The load at failure for n = 22 specimens was observed, and we found that x = 13.71 and s = 3.55.
We want to ind a tolerance interval for the load at failure that includes 90% of the values in the population with 95%

conidence. From Appendix Table XII, the tolerance factor k for n = 22, γ = 0.90, and 95% coni dence is k = 2.264.
The desired tolerance interval is
( x − ks x + ks)
,
or
. )
. )
⎡ ⎣ 13 71 −( 2 264 3 55 13 71 + ( 2 264 3 55⎤ ⎦
.
.
.
,
.
which reduces to (5.67, 21.74).
Practical Interpretation: We can be 95% conident that at least 90% of the values of load at failure for this particular

alloy lie between 5.67 and 21.74 megapascals.
From Appendix Table XII, we note that as n → ∞, the value of k goes to the z-value associ-
ated 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 = 1.645 as
0.05
n → ∞. Note that as n → ∞, a 100(1 – α)% prediction interval on a future value approaches a
tolerance interval that contains 100(1 – α)% of the distribution.
Exercises FOR SECTION 8-6

Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion

8-73. Consider the tire-testing data described in 8-79. Consider the television tube brightness test described
Exercise 8-29. Compute a 95% prediction interval on the life of the in Exercise 8-37. Compute a 99% prediction interval on the bright-
next tire of this type tested under conditions that are similar to those ness of the next tube tested. Compare the length of the prediction
employed in the original test. Compare the length of the prediction interval with the length of the 99% CI on the population mean.
interval with the length of the 95% CI on the population mean. 8-80. Consider the suspension rod diameter measure-
8-74. Consider the Izod impact test described in Exercise 8-30. ments described in Exercise 8-40. Compute a 95% prediction
Compute a 99% prediction interval on the impact strength of interval on the diameter of the next rod tested. Compare the
the next specimen of PVC pipe tested. Compare the length of length of the prediction interval with the length of the 95% CI
the prediction interval with the length of the 99% CI on the on the population mean.
population mean. 8-81. Consider the test on the compressive strength of
8-75. Consider the syrup-dispensing measurements concrete described in Exercise 8-39. Compute a 90% predic-
described in Exercise 8-31. Compute a 95% prediction interval tion interval on the next specimen of concrete tested.
on the syrup volume in the next beverage dispensed. Compare 8-82. Consider the bottle-wall thickness
the length of the prediction interval with the length of the 95% measurements described in Exercise 8-42. Compute a 90% pre-
CI on the population mean. diction interval on the wall thickness of the next bottle tested.
8-76. Consider the natural frequency of beams described in 8-83. Consider the fuel rod enrichment data described in
Exercise 8-34. Compute a 90% prediction interval on the diam- Exercise 8-43. Compute a 90% prediction interval on the enrich-
eter of the natural frequency of the next beam of this type that ment of the next rod tested. Compare the length of the prediction
will be tested. Compare the length of the prediction interval interval with the length of the 99% CI on the population mean.
with the length of the 90% CI on the population mean. 8-84. How would you obtain a one-sided prediction bound
8-77. Consider the rainfall in Exercise 8-35. Compute a on a future observation? Apply this procedure to obtain a 95%
95% prediction interval on the rainfall for the next year. Com- one-sided prediction bound on the wall thickness of the next
pare the length of the prediction interval with the length of the bottle for the situation described in Exercise 8-42.
95% CI on the population mean. 8-85. Consider the tire-testing data in Exercise 8-29. Compute
8-78. Consider the margarine test described in Exercise 8-38. a 95% tolerance interval on the life of the tires that has coni -
Compute a 99% prediction interval on the polyunsaturated dence level 95%. Compare the length of the tolerance interval
fatty acid in the next package of margarine that is tested. Com- with the length of the 95% CI on the population mean. Which
pare the length of the prediction interval with the length of the interval is shorter? Discuss the difference in interpretation of
99% CI on the population mean. these two intervals.