Hypothesis Testing

I haven't got through the whole chapter on Hypothesis testing yet so I may be jumping the gun but in Que 12.1, to get a Type II error, we reject H0 that mu is 20 and use mu is 30 - I don't quite get this sort of test as yet. If we reject that mu is 20, why would we assume it must be 30. I guess this comes down to why we made that the test in the first place and I know this is just a sample question. I get the concept of having a precise statement Ho:mu=20 and H1:mu<>20. I cant see what situation we would have H1:mu=30 though?

 

Hi

In general in order to compute the probability of Type II error, you need to compute under any given mu <> 20 i.e. a value for 'mu' belonging to the alternate hypothesis space as 'mu = 20' is the null hypothesis.

In this particular problem, the question states that the alternate hypothesis consists of only one value i.e. mu = 30. Hence the probability of Type II error is computed by setting mu = 30.

 

Right - so this test is saying there are only 2 possibilities mu=20 or mu=30 and it can't equal anything else?

 

Yes that would be the case for this particular example.

 

In that example we are given the test without reference to how we arrived at it. However, in Example 12.1, we decide upon the H1 to be mu>100 and end up concluding that it is likely the average IQ is greater than 100. Now I get that since we made H1 mu>100 rather than mu<>100, we used a one sided test. However if we made H1: mu<100 wouldn't we have done exactly the same test and concluded that the average IQ is less than 100?

 

Hi

It is a good question !

Remember when you are performing a hypothesis testing, you are only checking if your data is giving you sufficient evidence (at a given significance level) whether you can accept or reject null hypothesis only. However it is important you identify the alternate space correctly. (Refer to a similar discussion within the ActEd notes in section 2.2 of Chapter 12).

Now, in case you change the alternate to mu < 100, you reference critical point is the 0.05 (and not the 0.95) point of the Normal distribution i.e. -1.6445 as you are now looking at the left of the mean space. So, in this example as your test statistic value is 1.768, you will end up accepting Ho (albeit incorrectly !).

Had you actually tested against mu <> 100, your reference critical points would be -/+ 1.96. As the observed test statistic value lies between these points, you actually can't reject the null hypothesis at 5% significance !!

 

Right so if H1 is < we use the left tail as the critical region and > use the right tail of the critical region - I had missed that. Thanks. So, if we had wanted to test that IQs were <100 what test would we have set up here?

 

Same one I mentioned in my earlier reply.

Except that with the data given, we will not have sufficient evidence to reject Ho: mu = 100

 

Yes of course - so the test is valid. Its just useless. Thanks for your help.