Solution to question 4.3
A 100% confidence interval would, in general, be infinitely wide.
The only way to be 100% sure of an estimate of anything is to include all of the possible values that it could take.
There will be some cases (for non-normal data) where a 100% confidence interval would not be infinitely wide. If we consider the estimate of the mean number of teeth that a population has. We know that the number of teeth can't be less than zero so the lower end of a 100% confidence interval will be zero. The upper end of the 100% confidence interval would be 32 if we were sure that there were no genetic oddities in our population. As we can't be certain of this we would have to find out what the world record for number of teeth is and set this as our upper limit.
The following table shows the numbers we multiply the standard error by when calculating various confidence intervals. Notice how the numbers increase much more rapidly than the increase in confidence level. The case of 100% confidence requiring an infinite interval is the extreme end of this series.
| Size of confidence interval | Number of standard errors each side of the mean |
|---|---|
| 90% | 1·65 |
| 95% | 1·96 |
| 99% | 2·58 |
| 99·9% | 3·29 |
| 99·99% | 3·89 |
