Hypothesis Testing

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Homework Helper
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.

FactChecker
Gold Member
The two cases that you describe can not have a useful test. The cases of mean=0 versus mean=0.0000000000001 will not be distinguishable without billions of samples. You must start with an alternative hypothesis where a reasonable sample has a chance of being convincing. It is not usually necessary to determine the distribution associated with the alternative hypothesis. The assumption of the null hypothesis gives you the distribution that you will use and a sample that is out of line with that distribution allows you to convincingly state that the alternative hypothesis is the better choice.

CORRECTION: In your stated cases, if the sample mean is large enough, you can reject the null hypothesis.

Last edited:
WWGD and Stephen Tashi
Stephen Tashi