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AP Statistics Chapter 11 Notes 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. 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. 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. 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. 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) 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. 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 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. If the p-value is > .05, our result is not that unlikely to occur. Therefore we reject the null hypothesis. 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 pvalue if you choose the 2-sided option. 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. 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 badlyproduced data. 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. Errors Continued 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