Quiz #7

1. If Welch's confidence interval method is anticonservative in a certain situation it will generally produce confidence intervals which are narrower than they need to be.
2. If Welch's confidence interval method is anticonservative in a certain situation it will generally produce confidence intervals which are wider than they need to be.
3. If a hypothesis testing method is anticonservative in a certain situation it can be considered to be valid.
4. If a hypothesis testing method is anticonservative in a certain situation it cannot be considered to be valid.

• [ A ] 1 & 3
• [ B ] 1 & 4
• [ C ] 2 & 3
• [ D ] 2 & 4

• A confidence interval procedure that has a coverage probability less than the nominal level is anticonservative. If Welch's method produces wider confidence intervals than necessary in a certain situation, the coverage probability will exceed the nominal level (since a shorter interval could cover with the nominal coverage probability) and the procedure would be conservative. If the intervals are narrower than they need to be, the coverage probability will be too small, and the procedure would be anticonservative. So 1 is true and 2 is false. (The conservative attitude is that it is better to be safe than sorry --- if a failure to cover the true value is considered to be a horrible mistake, then by using a method that has a higher than stated success rate is offering an increased level of protection against failure ... being safe ... being conservative, as opposed to taking on an increased risk of failure (having a larger than desired chance of failure). Of course using a method for which the actual success rate corresponds to the nominal level may be what is desired, but often one has to choose between being conservative or being anticonservative.)
• A conservative test has a type I error rate less than the nominal level of the test, and so a rejection of the null hypothesis can be taken as serious evidence against the null hypothesis and in favor of the alternative hypothesis --- a rejection is still valid evidence against the null hypothesis. An anticonservative test is one for which the actual chance of a type I error can be larger than the nominal level of the test. Since false rejections of the null hypothesis can occur with too high of a probability, a rejection cannot be taken so seriously. If the nominal level of the test is 0.05 and the test is anticonservative and rejects with probability 0.2 when the null hypothesis is true, then a rejection just doesn't carry the same weight as it would if it could be guaranteed that a false rejection can only occur with a rather small probability --- we wouldn't have a valid test that is guaranteed to have a type I error probability no larger than 0.05. So 3 is false and 4 is true.