Type II error, also known as a "false negative" or "beta error," occurs in statistical hypothesis testing when a false null hypothesis is not rejected. In other words, it happens when a test fails to detect a significant effect or difference that truly exists in the population.

To understand Type II error, it's important to review the basic concepts of hypothesis testing:

• Null Hypothesis: A statement that there is no significant difference, effect, or relationship in the population. It represents the status quo or the default assumption.
• Alternative Hypothesis: A statement that contradicts the null hypothesis, suggesting the presence of a significant difference, effect, or relationship.
• Significance Level (α): The chosen probability threshold for rejecting the null hypothesis. Commonly used values include 0.05, 0.01, or 0.10.
• Power of the Test (1−β): The probability of correctly rejecting a false null hypothesis. It is the complement of the Type II error rate (β).

In a hypothesis test, the goal is to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis. A Type II error occurs when the null hypothesis is not rejected, even though it is false.

The probability of committing a Type II error is denoted by the symbol β. Mathematically, it can be expressed as:

P(Type II Error) = P(Fail to Reject Null Hypothesis | Alternative Hypothesis is True)

The power of a statistical test is defined as 1−β and represents the probability of correctly rejecting a false null hypothesis.

Common examples of Type II errors include:

• Medical Testing: Failing to diagnose a patient with a disease when, in fact, they have it (false negative).
• Criminal Justice: Failing to convict a guilty person (failing to reject the null hypothesis of innocence).
• Quality Control: Failing to identify a manufacturing process as defective when it is producing faulty products.

Researchers and analysts aim to minimize the risk of Type II errors by increasing the power of the test. This can be achieved by increasing the sample size, selecting a more sensitive test, or choosing a higher significance level (though this increases the risk of Type I errors). The trade-off between Type I and Type II errors is a crucial consideration in the design and interpretation of hypothesis tests.