Had the following question on my exam today and I'm just wondering if I did this correctly.
From a normally distributed population with mean and variance a sample has been drawn:
a) Test against on a 5% significance level under the assumption that σ2 is unknown.
c) Do the same test, given that .
Solution: I know that is to be rejected if , where T is the teststatistic and , where is the significance level.
So lets first compute them for a), where we need to use a t-distribution since σ is unknown, so we have that . Also, the estimator for is
so, . We also have that .
The teststatistic is
which means that we can not reject H0.
For b) we do an identical approach but wtith and we use a z-test instead. Here we have that The teststatistic in this case is
thus we reject .
Is this correct?
What my professor does in the solutions is he just makes a 95% confidenceinterval and just arrives to the same conclusions as I've done. Is his method correct without using the teststatistics?