Question

Explain the conceptual and quantitative relationships between Alpha risk and Beta risk when testing hypotheses, and include the impact sample size plays in managing these risk levels. This is a double-discussion assignment (80 points) that should be submitted as a Microsoft Word or Adobe Acrobat document submission. There are no responses to other students required (therefore, the assignment will close and lock shortly after the due date and time). Some students have been asking

Answer 1

Solution and Explanation:

The concept of Null hypothesis first.

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.

Alpha risk:This is the risk when we Reject Null when Null is true.

Beta risk: The risk when we Do NOT reject Null but Null is false

Impact of Sample size: 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.

$\alpha=P_{H_{0}}\left(\text {Reject} H_{0}\right)$ is the type 1 error.

$\beta=P_{H_{1}}\left(\text {RejectH}_{0}\right)$ is power

$1-\beta$ is the type 2 error

 $\alpha-\text {risk}=E(L(\theta, \delta))$

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.