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

Last updated date: 22nd Jun 2024
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Hint: We first express the concept of type 1 error and type 2 error probabilities. Then we state the theorems for calculating the probabilities.

Complete step by step solution:
When the null hypothesis $\left( {{H}_{0}}:\mu ={{\mu }_{0}} \right)$ is true and we reject it, we make a type 1 error. The probability of making a type I error is $\alpha $, which is the level of significance you set for your hypothesis test. An $\alpha $ of 0.05 indicates that we are willing to accept a 5% chance that we are wrong when you reject the null hypothesis.
When the null hypothesis is false and we fail to reject it, you make a type 2 error. The probability of making a type II error is $\beta $, which depends on the power of the test
The formulas for finding the probabilities are
Type 1: \[P\left( \text{rejecting }{{H}_{0}}|{{H}_{0}}\text{ true} \right)\] and Type 2: \[P\left( \text{accepting }{{H}_{0}}|{{H}_{0}}\text{ false} \right)\].

To lower this risk in type 1, we must use a lower value for $\alpha $. However, using a lower value for alpha means that we will be less likely to detect a true difference if one really exists. We can decrease your risk of committing a type 2 error by ensuring our test has enough power. We can do this by ensuring your sample size is large enough to detect a practical difference when one truly exists.