S2 Hypothesis Testing - Why 'as bad or worse'?

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peterw55
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I will preface this by apologising if this is a stupid question.

From the Edexcel textbook: The phrase 'as bad or worse' is sometimes helpful. We calculate the probability of getting evidence as bad or worse than that we have been presented with to make our judgement.

Why it is collective? Why do we calculate, for example, P(X<=1) rather than just P(X=1)?
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peterw55
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(Original post by petrus123)
I will preface this by apologising if this is a stupid question.

From the Edexcel textbook: The phrase 'as bad or worse' is sometimes helpful. We calculate the probability of getting evidence as bad or worse than that we have been presented with to make our judgement.

Why it is collective? Why do we calculate, for example, P(X<=1) rather than just P(X=1)?
Any help would be much appreciated.
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Gregorius
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(Original post by petrus123)
I will preface this by apologising if this is a stupid question.
On the contrary, this is an extraordinarily good question!

From the Edexcel textbook: The phrase 'as bad or worse' is sometimes helpful. We calculate the probability of getting evidence as bad or worse than that we have been presented with to make our judgement.

Why it is collective? Why do we calculate, for example, P(X<=1) rather than just P(X=1)?
This is one of the "dirty little secrets" of the so-called frequentist approach to statistics and one of the things that drives many statisticians towards the Bayesian school of statistics (as it bases inference on not "exactly what has been observed", but what "could have been observed").

The basic problem is that if you are dealing with continuous probability distributions, then P(X = a) = 0 for any singleton value a. Since you are basing your hypothesis test on an observed value a, and you really would like to come out with a meaningful probability statement about a being observed, this is a problem!

The trick that is used in frequentist statistics is to say that you want to calculate the probability of a value equal to, or more extreme than, a being observed. This is, to some extent, arbitrary. You could equally calculate the probability of a value being observed within any interval around a - but the question then arises about how you choose that interval and how you interpret the probabilities that pop out. So, that's the convention that has become established and what we're used to using.
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univ4464
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(Original post by petrus123)
I will preface this by apologising if this is a stupid question.

From the Edexcel textbook: The phrase 'as bad or worse' is sometimes helpful. We calculate the probability of getting evidence as bad or worse than that we have been presented with to make our judgement.

Why it is collective? Why do we calculate, for example, P(X<=1) rather than just P(X=1)?
I suppose as well, if you have a large discrete distribution, most values are going to be less than the significance level even though they are the same as / close to the expected outcome so it only makes sense to ask what is the probability that it lies in a particular region. Looking at it from the reverse, there are the critical regions that you say are unlikely enough to happen to discount the hypothesis, and in the hypothesis test you test to see whether the result is in that region.
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