Finding the beta risk from the alpha risk

[DivGradCurl]

Hello,
I would like to know the procedure in order to find the beta risk once the hypothesis test has been made.

I am aware of the fact that it is efficient to set both alpha and beta prior to data collection, but, in this case, I was given the observations and an alpha value. I used the t-test to compare the means, and one of the requirements is to test the hypothesis that the variances are the same. From the information that I have, the p-values and F-values do not allow the rejection of the null, which makes it possible to pool the variances and calculate the t-statistic for the first hypothesis. I understand how the alpha and beta risks are pictorially represented and what they mean, but unfortunately I don't see how to get this.

The answer to this question would be useful; it would be possible to state the risk of having assumed equal variances.

Any help is highly appreciated.

 

[EnumaElish]

If you have a critical value and a distribution (i.e. a mean and a variance) then you should be able to calculate both the "alpha" and the "beta" -- by which I am assuming you mean Type I and Type II errors. Can you explain a little?

Is this homework?

 

[DivGradCurl]

No. This is not homework. I'm using a computer program to analyze data. Just an alpha value (5%) was provided along with it. I am trying to find the procedure used to calculate the correspondent beta. Yes, alpha and beta stand for Type I and II errors, respectively. I can't just pick a value and say it is true. There must be some mathematical explanation (since the value was not predetermined).

Does this answer your question?

 

[EnumaElish]

Since you know alpha, you know the critical value: given the location (mean) and the spread (variance) of the distribution, the "z" value that equates the tail probability to your alpha value is the critical value z_c.

 

[EnumaElish]

Since you know z_c, all you have to do is to look at the other (alternative) distribution and calculate its tail probability, which would be the beta.

The other distribution is determined by the other (alternative) mean and the variance (since variances tested identical, you can assume the same variance for both distributions).