Day 16: STA205 - Introduction to Statistical Methods

Today:

Announcements:
- Quiz 8 (proportions)
Chapter 7: Testing conjectures about when the value of is known
- By an hypothesis, we will mean a conjecture about the nature of a population (usually concerning a parameter value).
- A test of hypothesis is a statistical procedure used to make a decision about the conjectured value of a population parameter.
- p-value (Definition 7.5, p. 237): the observed significance level (or p-value) of a test of hypothesis is the probability of obtaining the observed value of the sampled statistic, or a value of this statistic that is even more supportive of the alternative hypothesis, under the assumption that the null hypothesis is true.
- Please refer to your Hypothesis testing handout
  Note: there's a little error in your copy: the z-statistic should have an ${\bar{x}}$ in it, where I've written x.
  If you're really good, you caught that error!;)
- Example from our first test: Problem 5 can be phrased in this terminology. We were "testing a conjecture", but we can formalize the process as a test of hypothesis (see p. 240).
  1. Null hypothesis: ${\mu = 265}$
  2. Alternative hypothesis: ${\mu > 265}$
  3. Decision rule: ${p\le{.05}}$ , then reject the null in favor of the alternative.
  4. Test statistic: z-statistic, ${Z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}}$
  5. Compute p-value (value as extreme as Z=2.08 or more so): ${p = .0188}$ , hence reject the null in favor of the alternative.
  6. Interpretation of the results: older folks apparantly listen to more radio than other adults (our estimate is 290 hours per year).
- Examples:
  - From the course website of worked examples
    Warning: I don't like the wording of this example (or of our text): honestly, I'm so skittish that I'd prefer not to accept hypotheses: we either
    
    reject them "in favor of the alternative", or
    fail to reject them.
    I won't be bothered if you prefer the book's terminology, however.
  - #1, p. 242 (proportion)
Section 7.2: Performing a test of hypothesis about
- We'll use a "Z-Test" (see p. 246) when conditions are appropriate (see p. 245).
- In these problems, the final step is interpretation. There are two guides to interpretation, on pages 252 and 254.
- Examples:
  - Example 7.5 (also given on the course website of worked examples -- interesting twist on the sample size...)
  - #2, p. 256
  - #5, p. 257

Section 7.3: Possible errors and level of significance

Reality versus Decision: Type I and Type II errors

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

The following table (thanks Wikipedia) illustrates the situation:

		Actual condition
		Present	Absent
Test result	Positive	Condition Present + Positive result = True Positive	Condition absent + Positive result = False Positive Type I error
Test result	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.
Examples:
- #3, p. 265
- #4, p. 265

Links

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