Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Fall 2016 Room 150 Harvill Building 10:00 - 10:50 Mondays, Wednesdays & Fridays. Welcome http://today.msnbc.msn.com/id/33411196/ns/today-today_health/ http://www.youtube.com/watch?v=0r7NXEWpheg

(Veteran’s Day Holiday) Homework Assignments Homework Assignment #21 Hypothesis testing ANOVA: Analysis of Variance (More than two samples) Due Monday November 14th No Class on Friday (Veteran’s Day Holiday)

Before next exam (November 18th) Please read chapters 1 - 11 in OpenStax textbook Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence

Lab sessions Everyone will want to be enrolled in one of the lab sessions Labs continue this week Project 3 due & Start Project 4

This lab builds on the work we did in our very first lab This lab builds on the work we did in our very first lab. But now we are using the correlation for prediction. This is called regression analysis

What happened? We ran more subjects: Increased n So, we decreased variability Easier to find effect significant even though effect size didn’t change This is the sample size This is the sample size Small sample Big sample 8

What happened? We ran more subjects: Increased n So, we decreased variability Easier to find effect significant even though effect size didn’t change This is variance for each sample (Remember, variance is just standard deviation squared) This is variance for each sample (Remember, variance is just standard deviation squared) Small sample Big sample 9

Study Type 3: One-way Analysis of Variance (ANOVA) We are looking to compare two means Study Type 2: t-test Study Type 3: One-way Analysis of Variance (ANOVA) Comparing more than two means

“df” = degrees of freedom Remember, you should know these two formulas by heart “SS” = “Sum of Squares” “SS” = “Sum of Squares” = s2 = Sample Variance Sample Standard Deviation = s = “SS” = “Sum of Squares” “df” = degrees of freedom

Study Type 3: One-way ANOVA Single Independent Variable comparing more than two groups Single Dependent Variable (numerical/continuous) Used to test the effect of the IV on the DV Ian was interested in the effect of incentives for girl scouts on the number of cookies sold. He randomly assigned girl scouts into one of three groups. The three groups were given one of three incentives and looked to see who sold more cookies. The 3 incentives were 1) Trip to Hawaii, 2) New Bike or 3) Nothing. This is an example of a true experiment Dependent variable is always quantitative Sales per Girl scout Sales per Girl scout None New Bike Trip Hawaii None New Bike Trip Hawaii In an ANOVA, independent variable is qualitative (& more than two groups)

One-way ANOVA versus Chi Square Be careful you are not designing a Chi Square If this is just frequency you may have a problem This is a Chi Square Total Number of Boxes Sold Sales per Girl scout This is an ANOVA None New Bike Trip Hawaii None New Bike Trip Hawaii These are just frequencies These are just frequencies These are just frequencies These are means These are means These are means

One-way ANOVA One-way ANOVAs test only one independent variable Number of cookies sold One-way ANOVA None Bike Hawaii trip Incentives One-way ANOVAs test only one independent variable - although there may be many levels “Factor” = one independent variable “Level” = levels of the independent variable treatment condition groups “Main Effect” of independent variable = difference between levels Note: doesn’t tell you which specific levels (means) differ from each other A multi-factor experiment would be a multi-independent variables experiment

Comparing ANOVAs with t-tests Similarities still include: Using distributions to make decisions about common and rare events Using distributions to make inferences about whether to reject the null hypothesis or not The same 5 steps for testing an hypothesis Tells us generally about number of participants / observations Tells us generally about number of groups / levels of IV The three primary differences between t-tests and ANOVAS are: 1. ANOVAs can test more than two means 2. We are comparing sample means indirectly by comparing sample variances 3. We now will have two types of degrees of freedom t(16) = 3.0; p < 0.05 F(2, 15) = 3.0; p < 0.05 Tells us generally about number of participants / observations

A girl scout troop leader wondered whether providing an incentive to whomever sold the most girl scout cookies would have an effect on the number cookies sold. She provided a big incentive to one troop (trip to Hawaii), a lesser incentive to a second troop (bicycle), and no incentive to a third group, and then looked to see who sold more cookies. How many levels of the Independent Variable? What is Independent Variable? Troop 1 (nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 What is Dependent Variable? How many groups? n = 5 x = 10 n = 5 x = 12 n = 5 x = 14

Hypothesis testing: Step 1: Identify the research problem Is there a significant difference in the number of cookie boxes sold between the girlscout troops that were given the different levels of incentive? Describe the null and alternative hypotheses

Hypothesis testing: Decision rule = .05 Degrees of freedom (between) = number of groups - 1 = 3 - 1 = 2 Degrees of freedom (within) = # of scores - # of groups = (15-3) = 12* Critical F (2,12) = 3.98 *or = (5-1) + (5-1) + (5-1) = 12.

Appendix B.4 (pg.518) F (2,12) α= .05 Critical F(2,12) = 3.89 19

ANOVA table Source df MS F SS Between 40 ? ? ? ? Within 88 ? ? ? Total “SS” = “Sum of Squares” - will be given for exams - you can think of this as the numerator in a standard deviation formula ANOVA table Source df MS F SS Between 40 ? ? ? ? Within 88 ? ? ? Total 128 ? ?

Writing Assignment - ANOVA 1. Write formula for standard deviation of sample 2. Write formula for variance of sample 3. Re-write formula for variance of sample using the nicknames for the numerator and denominator SS df = MS 4. Complete this ANOVA table ANOVA table Source SS df MS F Between 40 ? ? ? Within 88 ? ? Total 128 ?

ANOVA table Source df MS F SS Between 40 ? 2 ? ? ? Within ? 88 12 ? ? “SS” = “Sum of Squares” - will be given for exams ANOVA table Source df MS F SS Between 40 ? 2 # groups - 1 ? ? ? 3-1=2 15-3=12 Within ? 88 12 ? ? # scores - number of groups Total 128 ? ? 14 # scores - 1 15- 1=14

“SS” = “Sum of Squares” - will be given for exams ANOVA table SSbetween dfbetween “SS” = “Sum of Squares” - will be given for exams 40 2 40 2 =20 MSbetween MSwithin ANOVA table Source df MS F SS 20 7.33 =2.73 Between 40 2 ? 20 ? 2.73 Within 88 12 7.33 ? Total 128 14 SSwithin dfwithin 88 12 =7.33 88 12

Make decision whether or not to reject null hypothesis Observed F = 2.73 Critical F(2,12) = 3.89 2.73 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis F(2,12) = 2.73; n.s. Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold The average number of cookies sold for three different incentives were compared. The mean number of cookie boxes sold for the “Hawaii” incentive was 14 , the mean number of cookies boxes sold for the “Bicycle” incentive was 12, and the mean number of cookies sold for the “No” incentive was 10. An ANOVA was conducted and there appears to be no significant difference in the number of cookies sold as a result of the different levels of incentive F(2, 12) = 2.73; n.s.

Thank you! See you next time!!