# N-P Lemma Test in Hypothesis testing and significance of inequality sign.

For the hypothesis test:

H_0 : mu= mu_0 (mu_1 > mu_0)

H_1 : mu= mu_1

Applying N-P Lemma: {X: lambda(mu_0,mu_1;X)>= k} Where lambda(mu_0,mu_1;X) is the ratio of likelihood functions; with the likelihood of alternate on the numerator and the likelihood of the null in the denominator.

-Essentially my question is: why use the sign >= ? Is this related to (mu_1 > mu_0) assumption?

-Secondly, I would like to confirm that the essential method (N-P Lemma) involves changing {X: lambda(mu_0,mu_1;X)>= k} into => {X: function of data only >= s} while keeping an eye on the inequality sign.

-Third: In order to get only the 'function of data' on one side, the parameters must be moved to the other side of inequality. Since parameters are not constants but have specific distributions; I assume 's' is then a function. However 's' is irrelevant as its size is determined by choosing an appropriate sized critical region. Is this correct?

Thank you (...and apologies for the poor notations, are there any beginner guides on using appropriate notations?)

-