Lecture Notes: Type 1 and Type 2 Errors in Statistics

Key Concepts

• Type 1 Error:

  • Occurs when the null hypothesis ((H_0)) is rejected when it is actually true.
  • Probability of this error is denoted by (\alpha).

• Type 2 Error:

  • Occurs when the null hypothesis is not rejected when it is actually false.
  • Probability of this error is denoted by (\beta).
  • The power of the test is (1 - \beta), which is the probability of correctly rejecting a false null hypothesis.

Decision Table

• Two possibilities for the null hypothesis ((H_0)):
  1. (H_0) is true
  2. (H_0) is false
• Two decision options:
  1. Reject (H_0)
  2. Fail to reject (H_0)
• Outcomes:
  • Type 1 Error: Reject (H_0) when true.
  • Correct Decision: Reject (H_0) when false.
  • Correct Decision: Fail to reject (H_0) when true.
  • Type 2 Error: Fail to reject (H_0) when false.

Example Problems

Example 1: John's Used Car

• Null Hypothesis ((H_0)): John's used car is safe to drive.

Identifying Errors:

• Type 1 Error: John thinks his car is not safe (rejects (H_0)) when it actually is safe.
• Type 2 Error: John thinks his car is safe (accepts (H_0)) when it is not safe.

Consequences:

• Type 2 Error Consequence: Greater consequence as it can lead to an accident if the car is unsafe.

Example 2: Criminal Court Case

• Null Hypothesis ((H_0)): Defendant is presumed innocent.

Identifying Errors:

• Type 1 Error: Jury finds defendant guilty (rejects (H_0)) when he is innocent.
• Type 2 Error: Jury finds defendant not guilty (accepts (H_0)) when he is not innocent.

Consequences:

• Type 1 Error Consequence: Greater consequence as an innocent person is wrongly punished.

Summary

• Understanding the differences between type 1 and type 2 errors is crucial for making informed decisions in hypothesis testing.
• Type 1 errors involve rejecting a true null hypothesis, while type 2 errors involve failing to reject a false null hypothesis.
• The consequences of these errors vary depending on the context and can significantly impact real-life decisions.