The formula to calculate the beta error (β) is:
\[ \beta = 1 - \alpha - (1 - \beta) \]
Where:
Beta error, also known as Type II error, occurs when a statistical test fails to reject a false null hypothesis. In other words, it is the error of not detecting an effect that is present. The probability of committing a Type II error is denoted by β. The power of a test, which is \( 1 - \beta \), represents the probability of correctly rejecting a false null hypothesis. Understanding and minimizing beta error is crucial in hypothesis testing to ensure the reliability and validity of the test results.
Definition: The standard error of beta measures the accuracy of the estimated beta coefficient in regression analysis.
Formula: \( SE(\beta) = \frac{S}{\sqrt{N}} \)
Example: \( SE(\beta) = \frac{10}{\sqrt{50}} \)
Definition: The standard error of the intercept (beta 0) in a regression model.
Formula: \( SE(\beta_0) = \frac{S}{\sqrt{N}} \)
Example: \( SE(\beta_0) = \frac{8}{\sqrt{40}} \)
Definition: The standard error of the slope (beta 1) in a regression model.
Formula: \( SE(\beta_1) = \frac{S}{\sqrt{N}} \)
Example: \( SE(\beta_1) = \frac{12}{\sqrt{60}} \)
Definition: Beta error (Type II error) occurs when a false null hypothesis is not rejected.
Formula: \( \beta = 1 - \text{Power} \)
Example: \( \beta = 1 - 0.8 \)
Definition: Alpha error (Type I error) occurs when a true null hypothesis is rejected, while beta error (Type II error) occurs when a false null hypothesis is not rejected.
Formula: \( \alpha = \text{Significance level} \)
Example: \( \alpha = 0.05 \)
