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## Summary:

- How to formalize the hypothesis of having a mean bigger than some value.

Hi, I have some set of data and I want to use Hypothesis Testing to discriminate between two hypotheses:

H0: My data follows a Gaussian distribution with a given mean and a given std (the actual values are ugly, so let's say mean = 0 and std = 1).

H1: My data follows a Gaussian distribution with mean > 0 and std = 1 (the same as before).

So, I want to use the maximum likelihood ratio to define my test statistic as

$$t(\vec{x})=\frac{f(\vec{x}|H_1)}{f(\vec{x}|H_0)}$$

So, for ##H_0## its clear that ##f(x|H_0)=N(0,1)##, but how do I find the expresion for ##f(x|H_1)?##.

Would be valid to compute the sample mean and, since it's actually bigger than 0, use $$f(x|H_1)=N(\bar{x},1)$$?

Thanks.

H0: My data follows a Gaussian distribution with a given mean and a given std (the actual values are ugly, so let's say mean = 0 and std = 1).

H1: My data follows a Gaussian distribution with mean > 0 and std = 1 (the same as before).

So, I want to use the maximum likelihood ratio to define my test statistic as

$$t(\vec{x})=\frac{f(\vec{x}|H_1)}{f(\vec{x}|H_0)}$$

So, for ##H_0## its clear that ##f(x|H_0)=N(0,1)##, but how do I find the expresion for ##f(x|H_1)?##.

Would be valid to compute the sample mean and, since it's actually bigger than 0, use $$f(x|H_1)=N(\bar{x},1)$$?

Thanks.