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Psyc 235: Introduction to Statistics http://www.psych.uiuc.edu/~jrfinley/p235/ DON’T FORGET TO SIGN IN FOR CREDIT! Stuff • Thursday: office hours hands-on help with specific problems • Next week labs: demonstrations of solving various types of hypothesis testing problems Population Sampling Distribution (of the mean) X Sample size = n X Descriptive vs Inferential • Descriptive describe the data you’ve got if those data are all you’re interested in, you’re done. • Inferential make inferences about population(s) of values (when you don’t/can’t have complete data) Inferential • Point Estimate • Confidence Interval • Hypothesis Testing 1 population parameter z, t tests 2 pop. parameters z, t tests on differences 3 or more?... ANOVA! Hypothesis Testing 1. Choose pop. parameter of interest • (ex: ) 2. Formulate null & alternative hypotheses • assume the null hyp. is true 3. Select test statistic (e.g., z, t) & form of sampling distribution • based on what’s known about the pop., & sample size Defining our hypothesis • H0= the Null hypothesis Usually designed to be the situation of no difference The hypothesis we test. • H1= the alternative hypothesis Usually the research related hypothesis Null Hypothesis (~Status Quo) Examples: • Average entering age is 28 (until shown different) • New product no different from old one (until shown better) • Experimental group is no different from control group (until shown different) • The accused is innocent (until shown guilty) Null Hypothesis Alternative Hypothesis H0 H1,H A ,H a H 0 : 0 H a : 0 H 0 : 0 ( 0 ) H 0 : 0 ( 0 ) H a : 0 H a : 0 - Ha is the hypothesis you are gathering evidence in support of. - H0 is the fallback option = the hypothesis you would like to reject. - Reject H0 only when there is lots of evidence against it. - A technicality: always include “=” in H0 - H0 (with = sign) is assumed in all mathematical calculations!!! Decision Tree for Hypothesis Testing Population Standard Deviation known? Yes Pop. Distribution normal? n large? (CLT) Yes No z-score Standard normal distribution Yes No No Test stat. Yes z-score Can’t do it t-score t distribution No Yes No t-score Can’t do it Selecting a distribution Hypothesis Testing 1. Choose pop. parameter of interest • (ex: ) 2. Formulate null & alternative hypotheses • assume the null hyp. is true 3. Select test statistic (e.g., z, t) & form of sampling distribution • based on what’s known about the pop., & sample size Hypothesis Testing 4. Calculate test stat.: sample stat. pop. param. under H std. dev, of sampling distribution 0 5. Note: The null hypothesis implies a certain sampling distribution 6. if test stat. is really unlikely under Ho, then reject Ho • HOW unlikely does it need to be? determined by Three equivalent methods of hypothesis testing (=significance level) Compute standardiz ed statistic. If standardiz ed statistic more extreme than critical value, then reject H 0 Compute 1 - confidence interval. If 0 not in confidence interval, then reject H 0 Compute p - value of observed X . If p - value , then reject H 0 p-value: prob of getting test stat at least as extreme if Ho really true. Hypothesis Testing as a Decision Problem H 0 true H 0 false H 0 retained Fail to Reject H 0 Great! Type II Error given H 0 false P(Type II Error) Power: Reject H 0 False Rejection of H 0 P(Type I Error) Significan ce Level Type I Error Great! Depends on sample size and how much the null and alternative hypotheses differ 1 – P(Type II error) Our ability to reject the null hypothesis when it is indeed false ERRORS • Type I errors (): rejecting the null hypothesis given that it is actually true; e.g., A court finding a person guilty of a crime that they did not actually commit. • Type II errors (): failing to reject the null hypothesis given that the alternative hypothesis is actually true; e.g., A court finding a person not guilty of a crime that they did actually commit. Type I and Type II errors decision criterion Power (1- -6 -4 -2 0 2 4 6 8 10 ANOVA: Analysis of Variance • a method of comparing 3 or more group means simultaneously to test whether the means of the corresponding populations are equal (why not just do a bunch of 2-sample ttests?...) inflation of Type I error rate ANOVA: 1-Way • You have sample data from several different groups • “One-way” refers to one factor. • Factor = a categorical variable that distinguishes the groups. • Level (group) of the factor refers to the different values that the categorical variable can take. ANOVA: 1-Way • Examples of Factors & groups: Factor: Political Affiliation groups: Democrat, Republican, Independent X=annual income Factor: Studying Method groups: Re-read notes, practice test, do nothing (control) X=score on exam ANOVA: 1-Way • So you’ve got 3(+) sets of sample data, from 3 different populations. • You want to test whether those 3 populations all have the same mean () • Null Hypothesis: H0: 1=2=3 (all pop. means are same) H1: all pop. means are NOT the same! • [draw examples on chalkboard] ANOVA: Assumptions • Normality populations are normally distributed • Homogeneity of variance populations have same variance (2) • 1-Way “Independent Samples”: groups are independent of each other ANOVA: the idea • Two ways to estimate 2 MSB: Mean Square Between Group (aka MSE: MS Error) based on how spread out the sample means are from each other. Variation Between Samples MSW: Mean Square Within Group based on the spread of data within each group Variation within Samples • If the 3(+) populations really do have same mean, then these 2 #s should be ~ the same • If NOT, then MSB should be bigger. ANOVA: calculating • MSB: Variation between samples (sample size) * (variance of sample means) if sample sizes are the same in all groups note: use the “sample variance” formula • MSW: Variation within samples (mean of sample variances) ANOVA: the F statistic • So how to compare MSB and MSW? Fdfn,dfd MSB MSW • Under H0: F≈1 • So calculate your F test statistic and compare to F distribution, see if it falls in region of rejection. [chalkboard] • note: F one-tailed! ANOVA: F & df • F distribution requires specification of 2 degrees of freedom values • DFn: degrees of freedom numerator: (# of groups) - 1 • DFd: degrees of freedom denominator: (total sample size (N)) - (# of groups) ANOVA: example • Groups: adults w/ 3 different activity levels • X=% REM sleep Group Very Active Moderately Active Inactive • • • • • Sample size 10 10 10 Sample mean 26.6 25.1 26.7 Sample variance 3 14.4 4.7 MSB=(sample size)(variance of sample means)=... MSW=(mean of sample variance)=... F=MSB/MSW=... dfn=# groups - 1=... dfd=Ntotal-#groups=... Fcritical=...p-value=...