The full quiz is here. The answers appear below. Comments, which are not part of the answers, are italicized.
Time: 20 minutes. Open book, open notes.
Each set of values below will be computed from the same random sample of an (assumed) Normal population. Put each set of values in order and provide a brief reason. If a unique order cannot be determined, state why not. If the values must be equal, say so.
1. (a) The 95% UCL of the mean; (b) the 99% UCL of the mean.
The 99% UCL is greater: to increase the coverage from 95% to 99%, you have to increase the upper limit.
2. (a) The 50% UCL of the mean; (b) the 50% LCL of the mean.
The UCL is always greater than the LCL.
3. (a) The 99%-content, 90%-confidence upper tolerance limit; (b) the 99% UCL of the 90th percentile.
An order cannot be determined. The first value will likely exceed 99% of the population and the second value will likely exceed 90% of the population. Therefore, for large sample sizes, the first value should exceed the second. However, the higher confidence demanded of the second value could cause this limit to be the larger, especially for small sample sizes. Table A.12c in Hahn & Meeker indicates the cross over in values occurs between sample sizes of 6 and 7. You can test this yourself using the Normal interval workbook.
4. (a) The 95% prediction interval for one future value; (b) the 95% prediction interval for all 95 future values.
The prediction interval must be wider to hold 95 future values. Therefore the lower limit (if any) of (b) will be lower and the upper limit (if any) will be higher than (a).
5. (a) The 90% prediction interval for four future values; (b) the 90% prediction interval for the mean of four future values.
The prediction interval (a) must be wider to hold all four values because their mean is much less variable.
6. (a) The 95% UCL of the 10th percentile; (b) the 95% LCL of the 90th percentile.
Most likely (a) is less than (b), but it is possible that the enormous uncertainty associated with low sample sizes (n=2 or 3) could reverse the order. (This does not actually happen, but it does happen when the confidence level exceeds 96.5% and n=2.)
7. (a) The 95% prediction limit for all 50 future values; (b) the 90%-content, 95%-confidence tolerance limit.
The prediction interval must contain all 50 values with 95% confidence, whereas the tolerance interval must contain 90% of the population with the same level of confidence. If it does, then we expect at least 45 of the next 50 values to fall within the tolerance interval, but we do not expect all 50 of the next 50 values to fall within the tolerance interval (at least not with the same high confidence). Therefore the prediction interval should be wider than the tolerance interval, regardless of sample size.
8. (a) The 99% UCL of the mean; (b) the 99% prediction limit for one future value.
The prediction limit must be higher to account for the variability in the mean and the variability in the future value.
Scoring: The passing score is 90 (seven out of eight correct answers).
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This page was created 28 March 2001.