How is the ratio of likelihoods in the Neyman Pearson lemma?

How is the ratio of likelihoods in the Neyman Pearson lemma?

The lemma tells us that, in order to be the most powerful test, the ratio of the likelihoods: should be small for sample points X inside the critical region C (“less than or equal to some constant k “) and large for sample points X outside of the critical region (“greater than or equal to some constant k “).

When do you use the nehman Pearson lemma?

Then, we can apply the Nehman Pearson Lemma when testing the simple null hypothesis H 0: μ = 3 against the simple alternative hypothesis H A: μ = 4. The lemma tells us that, in order to be the most powerful test, the ratio of the likelihoods:

Which is the best critical region of size α?

Then, C is a best critical region of size α if the power of the test at θ = θ a is the largest among all possible hypothesis tests. More formally, C is the best critical region of size α if, for every other critical region D of size α, we have:

How to test the nehman Pearson lemma for X?

Suppose X is a single observation (that’s one data point!) from a normal population with unknown mean μ and known standard deviation σ = 1 / 3. Then, we can apply the Nehman Pearson Lemma when testing the simple null hypothesis H 0: μ = 3 against the simple alternative hypothesis H A: μ = 4.

Which is the best Test of the null hypothesis h 0?

Consider the test of the simple null hypothesis H 0: θ = θ 0 against the simple alternative hypothesis H A: θ = θ a. Let C and D be critical regions of size α, that is, let: Then, C is a best critical region of size α if the power of the test at θ = θ a is the largest among all possible hypothesis tests.

Which is the p.d.f of a normal random variable?

Under the hypothesis H: μ = 12, the p.d.f. of a normal random variable is: for − ∞ < x < ∞ and σ > 0. In this case, the mean parameter μ = 12 is uniquely specified in the p.d.f., but the variance σ 2 is not.

How can we be sure that the t-test for a mean μ?

⌘ + ⇧ + F (Mac) As we learned from our work in the previous lesson, whenever we perform a hypothesis test, we should make sure that the test we are conducting has sufficient power to detect a meaningful difference from the null hypothesis. That said, how can we be sure that the T -test for a mean μ is the “most powerful” test we could use?