<u>Solution and Explanation:</u>
<u>The concept of Null hypothesis first.
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Null hypothesis is the prior belief about the parameter of interest. Normally, there ar good reasons for the belief, so we only change it (reject it ) if we have enough evidence to believe that it is wrong. Generally, this error should be small and to ensure that, we fix a small percentage of error bound for this. This is type 1 error and the corresponding risk is called alpha risk. The risk is the expected loss.
There are two types of error: Type 1: Reject Null when Null is true
Type 2: Do NOT reject Null when Null is false.
<u>Alpha risk:</u>This is the risk when we Reject Null when Null is true.
<u>Beta risk:</u> The risk when we Do NOT reject Null but Null is false
<u>Impact of Sample size:</u> increasing sample size controls the error. Larger the sample size is, the smaller the beta risk would be given a level of alpha risk(or equivalently type 1 error). Since we fix our type 1 error, we can only control beta risk. So increasing sample size improves beta risk as we get more information to make the decision.
is the type 1 error.
is power
is the type 2 error
where is the parameter. If is in the null region, the risk is alpha risk, when it is in the alternative region, the risk is beta risk.