Which one is correct about hypothesis testing?

[songoku]

Homework Statement

Which one is valid about hypothesis testing?
a) The distribution of test statistic under the null hypothesis is the same as its distribution across experiments using random samples of the same size from the same population

b) The test statistic has same distribution when the null hypothesis is true and when alternative hypothesis is true

c) The null hypothesis is always more likely to be correct than alternative hypothesis

d) We use the distribution of the test statistic when null hypothesis is true

e) The distribution of test statistic under the null hypothesis depends on the direction of the extreme which comes from the alternative hypothesis

Relevant Equations

None

My attempt:

a) I am not really sure I understand this option fully but my answer will be this one is only applicable if the sample size is large and central limit theorem can be applied so (a) is wrong

b) The test statistic has "same distribution" to what? My opinion is (b) is wrong because distribution of test statistic is independent of whether null and alternative hypothesis are correct or not

c) I think this is correct because the probability of alternative hypothesis to be correct is the same as significance level used which is always lower than probability to accept null hypothesis

d) This is wrong because of the same reasoning as (b), distribution of test statistic is independent of whether null hypothesis is correct or not

e) This is wrong because distribution of test statistic is independent of alternative hypothesis


Am I correct? Thanks

 

[WWGD]

It may differ depending on the statistic you're testing for: mean, proportion, F-test for equality of Variance, etc.

 

[songoku]

It may differ depending on the statistic you're testing for: mean, proportion, F-test for equality of Variance, etc.

Oh I don't know about that.

Let say the context is hypothesis testing for difference in mean

 

[FactChecker]

a) I am not really sure I understand this option fully but my answer will be this one is only applicable if the sample size is large and central limit theorem can be applied so (a) is wrong

I agree that it is false, but for a different reason. I interpret "under the null hypothesis" to mean that the null hypothesis is true and the sample is from that distribution. But if the null hypothesis is false, that is not true.

b) The test statistic has "same distribution" to what? My opinion is (b) is wrong because distribution of test statistic is independent of whether null and alternative hypothesis are correct or not

Again, I agree that it is false, but for a different reason. The null hypothesis and the alternative hypothesis usually hypothesize different distributions. So which hypothesis is true definitely effects the distribution and the associated test results.

c) I think this is correct because the probability of alternative hypothesis to be correct is the same as significance level used which is always lower than probability to accept null hypothesis

I disagree. IF the null hypothesis is true, the test result should end up agreeing with the null hypothesis a large percent of the time (95%, 99.5%, etc.). But the truth of the null hypothesis is not a "given". That is what you are testing. For example, suppose I am tossing a coin and the null hypotheses is that the coin is fair. Now suppose that the coin used is actually two-headed. Then the null hypothesis will not be accepted.

d) This is wrong because of the same reasoning as (b), distribution of test statistic is independent of whether null hypothesis is correct or not

We need to be careful here. The actual distribution of the test statistic is unknown. I interpret "we use" to mean "we hypothesize". We hypothesize the distribution of the test statistic to be the one derived if the null hypothesis is true.

e) This is wrong because distribution of test statistic is independent of alternative hypothesis

The hypothesized distribution of the test statistic assumes the null hypothesis is true. Then we test an alternative hypothesis, which will be accepted in one of three possible problem situations:
1) if the test result is too high (one-tail on the high side),
2) too low (one tail on the low side), or
3) too far from the hypothesized test statistic mean (two tails).