Day 9: STA205 - Introduction to Statistical Methods

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• Announcements:
  • Reminder: please hide your rectangles.

  • Your quiz is returned.

    We need to have a clear picture of "the story", and too many of you don't. Let me tell a story....

  • Your first test is next week, Thursday. It will cover everything through Packet 5.

  • Kylie Zent sent me the link to this Sampling Distribution of the Sample Proportion Calculator which she has found useful for checking her work.

• Today
  • We wrap up Packet 04 (Hypothesis Test for p). Today we're going to be talking about errors in hypothesis testing, because sometimes we make mistakes while testing:
    • 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 do not have AIDS (condition absent), the first error above is called a Type II error, while the second error is a Type I:

      • 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.

      So let's continue with Packet 04 (Hypothesis Test for p).

  • Then we'll begin Packet 5, which has two major objectives:
    1. estimate parameter $p$ using statistic $\hat{p}$, with given confidence, and
    2. discover the sample size needed to estimate $p$ with given confidence.

• Links:

  • Type I and Type II errors

  • Sampling Distribution of the Sample Proportion Calculator

  • The Standard Normal Table of probabilities.