Day 9: STA205 - Introduction to Statistical Methods

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• Announcements:
  • Reminder: stash your rectangles.

  • You have a quiz today.

  • You have a new IMath assignment, due Tuesday at 5:00 pm.

  • Today:
    • Science: we don't do truth, and we don't do proof: we test conjectures, hypotheses, theories. Sometimes they're about beer!

    • Today we're going to be talking about errors in hypothesis testing, because sometimes we make mistakes while testing:

      Type I and Type II errors

    • Alternative vocabulary: false positive, false negative....

      The following table (thanks Wikipedia) illustrates the situation in a typical medical context. Imagine that you're testing for AIDS. You either have it, or you don't. Now, suppose that you're tested for AIDS. The test either says you have it, or it doesn't. There are two kinds of errors the test can make:

      • You have AIDS, but the test says you don't.
      • You don't have AIDS, but the test says you do.
      That second error is bad, but the first one is worse!

      If the null hypothesis is that you have AIDS, the first error is called a Type I error, while the second error is a Type II:

      • Type I error: rejecting a true null hypothesis.
      • Type II error: failing to reject a false null hypothesis. (mnemonic: Type II, failing to)
      It's good to realize that we're getting two cases right!

      		Actual condition
      		Present	Absent
      Test  result	Positive	Condition Present + Positive result = True Positive	Condition absent + Positive result = False Positive Type I error
      	Negative	Condition present + Negative result = False (invalid) Negative Type II error	Condition absent + Negative result = True (accurate) Negative

    • The probability of a Type I error is given by $\alpha$, the level of significance of the test.

    • The probability of a Type II error is given by $\beta$

    • Obviously we'd like both types of errors to be small, but making one smaller tends to make the other larger! There's always a trade-off.

    • Packet 4: Hypothesis Test for Proportions

• Links:
  • Please watch Brooke's Hypothesis test for the proportion at some point.
  • The Standard Normal Table of probabilities.
  • Interview with Richard Feynman