AP Statistics

1. AP Statistics Chapter 11 Notes

2. Significance Test & Hypothesis • Significance test: a formal procedure for comparing observed data with a hypothesis whose truth we want to assess. • Hypothesis: a statement about a population parameter.

3. Null (Ho) and Alternative (Ha) Hypotheses • The null hypothesis is the statement being tested in a significance test. • Usually a statement of “no effect”, “no difference”, or no change from historical values. • The significance test is designed to assess the strength of evidence against the null hypothesis. • The alternative hypothesis is the claim about the population that we are trying to find evidence for.

4. Example: One-sided test • Administrators suspect that the weight of the high school male students is increasing. They take an SRS of male seniors and weigh them. A large study conducted years ago found that the average male senior weighed 163 lbs. • What are the null and alternative hypotheses? • Ho: μ = 163 lbs. • Ha: μ > 163 lbs.

5. Example: Two-sided test • How well do students like block scheduling? Students were given satisfaction surveys about the traditional and block schedules and the block score was subtracted from the traditional score. • What are the null and alternative hypotheses? • Ho: μ = 0 • Ha: μ ≠ 0 • *You must pick the type of test you want to do before you look at the data.* • Be sure to define the parameter.

6. Conditions for Significance Tests • SRS • Normality (of the sampling distribution) • For means: • 1. population is Normal or • 2. Central Limit Theorem (n > 30) or • 3. sample data is free from outliers or strong skew • For proportions: • np > 10, n(1 - p) > 10 • Independence (N > 10n)

7. Test Statistic • Compares the parameter stated in Ho with the estimate obtained from the sample. • Estimates that are far from the parameter give evidence against Ho. • For now we’ll us the z-test.

8. P-Value • Assuming that H0 is true, the probablility that the observed outcome (or a more extreme outcome) would occur is called the p-value of the test. • Small p-value = strong evidence against H0. • How small does the p-value need to be? • We compare it with a significance level (α – level) chosen beforehand. • Most commonly α = .05

9. P-value continued • If the p-value is as small or smaller than α, then the data are “statistically significant at level α”. • Ex: α = .05 • If the p-value is < .05, then there is less than a 5% chance of obtaining this particular sample estimate if H0 is true. • Therefore we reject the null hypothesis. • If the p-value is > .05, our result is not that unlikely to occur. • Therefore we fail to reject the null hypothesis. • If done by hand, the p-value must be doubled when performing a 2-sided test. The calculator will already display this doubled p-value if you choose the 2-sided option.

10. Confidence vs. Significance • Performing a level α 2-sided significance test is the same as performing a 1 – α confidence interval and seeing if μ0 falls outside of the interval. • e.g. If a 99% CI estimated a mean to be (4.27, 5.12), then a significance test testing the null hypothesis H0:µ = 4 would be significant at α = .01.

11. Reminders about Significance Tests • 1. Don’t place too much importance on “statistically significant”. • Smaller p-value = stronger evidence against H0 • 2.Statistical significance is not the same as practical importance. • 3. Don’t automatically use a test…examine the data and check the conditions. • 4. Statistical inference is not valid for badly-produced data.

12. Mistakes in significance testing • Type I error: • Reject H0 when H0 is actually true. • Type II Error: • Fail to reject H0 when H0 is actually false.

13. Errors Continued

14. Errors continued • The significance level α is the probability of making a Type I error. • Power: The probability that a fixed level α significance test will reject H0 when a particular alternative value of the parameter is true. • Ways to increase the power. • Increase α • Decrease σ • Increase n