How do you calculate type 1 error and type 2 error probabilities?

Type 1 error and Type 2 error are terms commonly used in hypothesis testing and statistical hypothesis testing. They are associated with the concepts of significance level (alpha, α) and power (1 - beta, β) in hypothesis testing. Here's how you calculate these probabilities:

Type 1 Error (False Positive, α):

Type 1 error occurs when you reject a null hypothesis that is actually true.
The probability of making a Type 1 error is denoted as α (alpha), and it is also known as the significance level.
α is typically set by the researcher before conducting a hypothesis test. Common values for α are 0.05 (5%) or 0.01 (1%).
To calculate α, you need to specify your chosen significance level and then determine the critical region of your test statistic distribution that corresponds to that level. If your test statistic falls into this critical region, you reject the null hypothesis and commit a Type 1 error with a probability of α.
Type 2 Error (False Negative, β):

Type 2 error occurs when you fail to reject a null hypothesis that is actually false.
The probability of making a Type 2 error is denoted as β (beta).
β depends on the power of the test, which is the probability of correctly rejecting a false null hypothesis. Power is denoted as (1 - β).
Calculating β can be more complex and often requires information about the true effect size, the sample size, and the specific statistical test being used.
In most practical situations, you may not have a specific value for β but can calculate it using software or statistical tables if you have the necessary information.
To summarize:

Type 1 error (α): The probability of incorrectly rejecting a true null hypothesis, typically set by the researcher.
Type 2 error (β): The probability of failing to reject a false null hypothesis, dependent on the power of the test (1 - β).
It's important to note that there is often a trade-off between Type 1 and Type 2 errors. Decreasing the probability of one type of error (e.g., reducing α to decrease Type 1 error) may increase the probability of the other type of error (e.g., increasing β and Type 2 error). Researchers must carefully consider the balance between these two error types when designing hypothesis tests.