Finding the beta risk from the alpha risk

  • Thread starter DivGradCurl
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In summary: So, the beta would be equal to the z-value (critical value) multiplied by the variance of the other distribution.
  • #1
DivGradCurl
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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.
 
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  • #2
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?
 
  • #3
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?
 
  • #4
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 zc.
 
  • #5
Since you know zc, 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).
 

1. What is the difference between beta risk and alpha risk?

Beta risk, also known as Type II error, is the probability of rejecting the null hypothesis when it is actually true. On the other hand, alpha risk, also known as Type I error, is the probability of accepting the null hypothesis when it is actually false. In simpler terms, beta risk is the probability of a false negative result, while alpha risk is the probability of a false positive result.

2. How do you calculate beta risk from alpha risk?

To calculate beta risk from alpha risk, you need to know the power of the statistical test. Power is the probability of rejecting the null hypothesis when it is actually false. Beta risk can be calculated by subtracting the power from 1 and then multiplying it by the alpha risk. This formula can be represented as (1 - Power) * Alpha risk = Beta risk.

3. Why is it important to find the beta risk from alpha risk?

It is important to find the beta risk from alpha risk because it helps us understand the accuracy and reliability of our statistical results. If the beta risk is high, it means that there is a high chance of missing an effect that actually exists, which can lead to incorrect conclusions. Therefore, by knowing the beta risk, we can better interpret the results of our statistical tests.

4. What factors can affect the beta risk from alpha risk?

The main factor that affects the beta risk from alpha risk is the sample size. A smaller sample size can increase the beta risk as it reduces the power of the statistical test. Other factors that can affect beta risk include the type of statistical test used, the significance level, and the variability of the data.

5. How can we reduce the beta risk from alpha risk?

There are a few ways to reduce the beta risk from alpha risk. One way is to increase the sample size, which can improve the power of the statistical test and decrease the beta risk. Another way is to use a more efficient statistical test with a higher power. Additionally, setting a lower significance level can also help reduce the beta risk, but this may increase the chance of making a Type I error.

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