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Welcome to Week 09 College Statistics http://media.dcnews.ro/image/201109/w670/statistics.jpg Hypothesis Tests So far, we have tested: Does µ = a value (Confidence Intervals) Hypothesis Tests Today we test whether means from different samples are different No Difference? Difference Between Means Remember the guy who drew the “leptokutic/platykurtic” pics? Difference Between Means William Sealy Gosset Difference Between Means Comparisons of means of two groups are called “t-Tests” Difference Between Means There are a variety of t-Tests Tests between two unrelated groups, each with its own treatment Tests between two related (paired) groups, each with its own treatment Difference Between Means To compare two paired group's means, use a: “Paired t-test” Difference Between Means “Paired”: two measurements on the same person measurements on twins two measurements for the same date, etc Difference Between Means We have two batches of stout: F5_10 and N25_5 Tasters are tasting each to see if they are the same Mr Gosset thinks they are not! Difference Between Means These are paired data because each analyst tasted each of the batches – two measurements BY the same person Difference Between Means H0: (what we want to disprove) There is no difference in the flavor of the two batches Ha: (what we want to prove) There is a difference in the flavor of the two batches Difference Between Means We hope to disprove H0 and thereby to prove Ha Paired t-Test PROJECT QUESTION Step 1: Paired t-Test PROJECT QUESTION Step 1: Set the α-level Paired t-Test PROJECT QUESTION Step 1: α-level = 0.05 Step 2: Paired t-Test PROJECT QUESTION Step 1: α-level = 0.05 Step 2: Set the practicallysignificant difference Paired t-Test PROJECT QUESTION Step 1: α-level = 0.05 Step 2: practically-significant difference = Mr Gosset says: 2 rating points Step 3: Paired t-Test PROJECT QUESTION Step 1: α-level = 0.05 Step 2: practically-significant difference = 2 rating levels Step 3: Set Ha (they are different) Ha: μF5_10 ≠ μN25_5 Step 4: Paired t-Test PROJECT QUESTION Step 1: α-level = 0.05 Step 2: practically-significant difference = 2 rating levels Step 3: Ha: μF5_10 ≠ μN25_5 Step 4: Set H0 Paired t-Test PROJECT QUESTION Step 1: α-level = 0.05 Step 2: practically-significant difference = 2 rating levels Step 3: Ha: μF5_10 ≠ μN25_5 Step 4: Set H0 (they are the same) H0: μF5_10 - μN25_5 = 0 Paired t-Test PROJECT QUESTION Step 1: α-level = 0.05 Step 2: practically-significant difference = 2 rating levels Step 3: Ha: μF5_10 ≠ μN25_5 Step 4: H0: μF5_10 - μN25_5 = 0 Difference Between Means Because each taster tastes each of the two batches, the data are paired Difference Between Means use: Data/ Data Analysis/ t-Test: Paired Two Sample for Means POOF! OK… now what? Difference Between Means If you had any reason to believe that one batch would be better than the other you would use: P(T<=t) one tail Difference Between Means We didn’t, so you would use: P(T<=t) two tail Difference Between Means The probability is .03 Will you reject H0? Difference Between Means How all the inferential tests work: Difference Between Means How all the inferential tests work: Excel calculates a probability that you would get the data you got if the null hypothesis were true Difference Between Means How all the inferential tests work: Excel calculates a probability that you would get the data you got if the null hypothesis were true If it’s ≤ α-level, reject H0 Paired t-Test PROJECT QUESTION The probability is .03 Do you reject H0? Paired t-Test PROJECT QUESTION The probability is .03 Do you reject H0? Yup! Difference Between Means CELEBRATE! Difference Between Means Remember – Reject the hypothesis if the statistic is smaller than 0.05 Paired t-Test PROJECT QUESTION Reject H0 ! Conclusion? Paired t-Test PROJECT QUESTION Reject H0 ! Conclusion: The batches do taste different! Paired t-Test PROJECT QUESTION Which tastes better? Paired t-Test PROJECT QUESTION For tests of differences between means, what would happen if you had a bigger sample size? Difference Between Means Demo of sample size in t-test Difference Between Means The P comes from a standardized t-distribution: Questions? Difference Between Means We use the t-test when we have paired data because it is more powerful We could use it for other twogroup comparisons, but we usually use another analysis: Difference Between Means Comparing Several Group's Means: “ANOVA” “Analysis of Variance” Difference Between Means t-tests can only be used for comparing two groups ANOVA can be used to compare two or more groups Difference Between Means A paired t-test is more powerful A non-paired t-test is THE SAME as an ANOVA (ANOVA’s Excel output page is better) Difference Between Means “ANOVA” stands for “ANalysis Of VAriance” Difference Between Means The analysis assigns the variability in the data to: the difference between the groups the difference between individuals Difference Between Means Sir Ronald Aylmer Fisher Difference Between Means Salaries for Criminal Justice Jobs Difference Between Means There are four classifications of jobs: probation, administration, correctional and patrol We want to compare the average salaries to see if they are the same ANOVA PROJECT QUESTION What would be a good value for α? ANOVA PROJECT QUESTION α = .05 What would be a good level of practical significance? ANOVA PROJECT QUESTION What is Ha? ANOVA PROJECT QUESTION Alternative hypothesis Ha: There are differences in the salaries of the four job classifications: μprobation ≠ μadministration ≠ μcorrectional ≠ μpatrol What is H0? ANOVA PROJECT QUESTION Null (no difference) hypothesis H0: There is no difference in salaries for the four job classifications: μprobation = μadministration = μcorrectional = μpatrol Difference Between Means Our strategy: We hope to disprove H0 and thereby to prove Ha ANOVA PROJECT QUESTION Why can’t you use a t-test for this data? Difference Between Means use: Data Data Analysis ANOVA: Single Factor Difference Between Means The first output table is just descriptive statistics: Difference Between Means The second table is the ANOVA table … EEK! Difference Between Means Don’t panic! Just look at the P-value Difference Between Means The P-value is the likelihood of our pattern of differences in the means IF H0 was TRUE Difference Between Means The probability is 1.28 E-5 or 0.0000128 (0.00128%) Is that very likely? ANOVA PROJECT QUESTION We said we would reject H0 if it was only 0.05 (5%) likely to be true Can we reject H0? Difference Between Means Remember – Reject the null hypothesis if the statistic is smaller than 0.05 Difference Between Means YES! If we are willing to be wrong in rejecting H0 5% of the time, 0.00128% is a whole lot less likely to be wrong! ANOVA PROJECT QUESTION What is your conclusion? Difference Between Means We conclude there is a significant difference between the average pay of the CJ job categories Difference Between Means Remember – Reject the null hypothesis if the statistic is smaller than 0.05 Questions? Difference Between Means For the CJ job classifications, we rejected H0 and concluded the salaries are different Difference Between Means But… Are they all different, or is just one different or two or … Difference Between Means Do a HI-Lo-Close Confidence Interval graph! Difference Between Means Difference Between Means Difference Between Means ANOVA PROJECT QUESTION Which means are significantly different? ANOVA PROJECT QUESTION Is the difference practically significant? Difference Between Means Now do you see why we’ve been doing Hi-Lo-Close graphs once a week since we learned them? Difference Between Means BTW: pre-Excel, this comparison used to be REALLY hard to do! Difference Between Means Yay Excel! Difference Between Means PROJECT QUESTION What would happen if you had a bigger sample size? Difference Between Means PROJECT QUESTION What would happen if you had a bigger sample size? You would be able to show more statistically significant differences Difference Between Means PROJECT QUESTION Difference Between Means t-tests and ANOVAs are designed to be VERY powerful for small sample sizes Difference Between Means That’s why we include a level of practical significance Difference Between Means Similar to previous tests, the P comes from a standardized F-distribution: Difference Between Means Because “z” and “t” are based on 𝒙 , they have similar shapes F is based on a variance, so it is in squared units! Questions?