Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2017 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays. Welcome http://www.youtube.com/watch?v=oSQJP40PcGI http://www.youtube.com/watch?v=oSQJP40PcGI

A note on doodling

By the end of lecture today 3/27/17 Hypothesis testing with z scores Hypothesis testing with t-tests Hypothesis testing with ANOVA

Before next exam (April 7th) 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

Homework Assignment 20 Please complete this homework worksheet two-sample t-tests Due: Wednesday, March 29th

Lab sessions Everyone will want to be enrolled in one of the lab sessions Project 3 this week

Prep Project 3 Study Type 2: t-test 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 Prep Project 3

Comparing ANOVAs with t-tests Prep Project 3 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

Prep Project 3 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 Prep Project 3

ANOVA: Using MS Excel Prep Project 3 A girlscout troop leader wondered whether providing an incentive to whomever sold the most girlscout 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. Troop 1 (Nada) 10 8 12 7 13 Troop 2 (bicycle) 12 14 10 11 13 Troop 3 (Hawaii) 14 9 19 13 15 Prep Project 3 n = 5 x = 10 n = 5 x = 12 n = 5 x = 14

Let’s do one Replication of study (new data) Prep Project 3

Let’s do same problem Using MS Excel Prep Project 3

Let’s do same problem Using MS Excel Prep Project 3

# scores - number of groups SSbetween dfbetween 40 2 40 2 =20 3-1=2 # groups - 1 MSbetween MSwithin # scores - number of groups 15-3=12 Prep Project 3 SSwithin dfwithin 88 12 =7.33 20 7.33 =2.73 88 12 # scores - 1 15- 1=14

Prep Project 3 No, so it is not significant Do not reject null F critical (is observed F greater than critical F?) P-value (is it less than .05?) Prep Project 3

Prep Project 3 Make decision whether or not to reject null hypothesis Observed F = 2.73 Critical F(2,12) = 3.89 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis Also p-value is not smaller than 0.05 so we do not reject the null hypothesis Step 6: Conclusion: There appears to be no effect of type of incentive on number of girl scout cookies sold Prep Project 3

Prep Project 3 Make decision whether or not to reject null hypothesis Observed F = 2.72727272 F(2,12) = 2.73; n.s. Critical F(2,12) = 3.88529 2.7 is not farther out on the curve than 3.89 so, we do not reject the null hypothesis 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. Prep Project 3

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

Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule Alpha level? (α = .05 or .01)? Critical statistic (e.g. z or t) value? Step 3: Calculations Step 4: Make decision whether or not to reject null hypothesis If observed z (or t) is bigger then critical z (or t) then reject null Step 5: Conclusion - tie findings back in to research problem

Hypothesis testing with t-tests The result is “statistically significant” if: the observed statistic is larger than the critical statistic observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x2) to be big!! the p value is less than 0.05 (which is our alpha) p < 0.05 If we want to reject the null, we want our “p” to be small!! we reject the null hypothesis then we have support for our alternative hypothesis Review

Independent samples t-test Are the two means significantly different from each other, or is the difference just due to chance? Independent samples t-test Donald is a consultant and leads training sessions. As part of his training sessions, he provides the students with breakfast. He has noticed that when he provides a full breakfast people seem to learn better than when he provides just a small meal (donuts and muffins). So, he put his hunch to the test. He had two classes, both with three people enrolled. The one group was given a big meal and the other group was given only a small meal. He then compared their test performance at the end of the day. Please test with an alpha = .05 Big Meal 22 25 Small meal 19 23 21 Mean= 21 Mean= 24 Got to figure this part out: We want to average from 2 samples - Call it “pooled” x1 – x2 t = 24 – 21 variability t = variability 22

α = .05 Independent samples t-test Step 1: Identify the research problem Did the size of the meal affect the learning / test scores? Step 2: Describe the null and alternative hypotheses Step 3: Decision rule α = .05 Two tailed test n1 = 3; n2 = 3 Degrees of freedom total (df total) = (n1 - 1) + (n2 – 1) = (3 - 1) + (3 – 1) = 4 Critical t(4) = 2.776 Step 4: Calculate observed t score 23

Notice: Simple Average = 3.5 Mean= 21 Mean= 24 Big Meal Deviation From mean -2 1 Small Meal Deviation From mean -2 2 Squared deviation 4 1 Squared Deviation 4 Big Meal 22 25 Small meal 19 23 21 Σ = 6 Σ = 8 6 3 Notice: s2 = 3.0 1 2 1 Notice: Simple Average = 3.5 8 4 Notice: s2 = 4.0 2 2 2 S2pooled = (n1 – 1) s12 + (n2 – 1) s22 n1 + n2 - 2 S2pooled = (3 – 1) (3) + (3 – 1) (4) 31 + 32 - 2 = 3.5 24

S2p = 3.5 Mean= 21 Mean= 24 Big Meal Deviation From mean -2 1 Small Meal Deviation From mean -2 2 Squared deviation 4 1 Squared Deviation 4 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 Σ = 6 Σ = 8 = 24 – 21 1.5275 24 - 21 = 1.964 3.5 3.5 3 3 Observed t Observed t = 1.964 Critical t = 2.776 1.964 is not larger than 2.776 so, we do not reject the null hypothesis t(4) = 1.964; n.s. Conclusion: There appears to be no difference between the groups 25

Type of test with degrees of freedom Value of observed statistic We compared test scores for large and small meals. The mean test scores for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there appears to be no significant difference in test scores between the two types of meals, t(4) = 1.964; n.s. Type of test with degrees of freedom n.s. = “not significant” p<0.05 = “significant” Value of observed statistic Start summary with two means (based on DV) for two levels of the IV Finish with statistical summary t(4) = 1.96; ns Describe type of test (t-test versus anova) with brief overview of results Or if it *were* significant: t(9) = 3.93; p < 0.05 26

Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 27

Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 28

Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 If checked you’ll want to include the labels in your variable range If checked, you’ll want to include the labels in your variable range If checked you’ll want to include the labels in your variable range 29

Complete a t-test Finding Means Finding Means 30

Complete a t-test This is variance for each sample (Remember, variance is just standard deviation squared) Please note: “Pooled variance” is just like the average of the two sample variances, so notice that the average of 3 and 4 is 3.5 31

Complete a t-test This is “n” for each sample (Remember, “n” is just number of observations for each sample) This is “n” for each sample (Remember, “n” is just number of observations for each sample) Remember, “degrees of freedom” is just (n-1) for each sample. So for sample 1: n-1 =3-1 = 2 And for sample 2: n-1=2-1 = 2 Then, df = 2+2=4 df = “degrees of freedom” 32

Finding degrees of freedom Complete a t-test Finding degrees of freedom 33

Complete a t-test Finding Observed t 34

Complete a t-test Finding Critical t 35

Finding Critical t 36

Finding p value (Is it less than .05?) Complete a t-test Finding p value (Is it less than .05?) 37

Step 4: Make decision whether or not to reject null hypothesis Complete a t-test Step 4: Make decision whether or not to reject null hypothesis Reject when: observed stat > critical stat 1.96396 is not bigger than 2.776 “p” is less than 0.05 (or whatever alpha is) p = 0.121 is not less than 0.05 Step 5: Conclusion - tie findings back in to research problem There was no significant difference, there is no evidence that size of meal affected test scores 38

Type of test with degrees of freedom Value of observed statistic We compared test scores for large and small meals. The mean test scores for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there appears to be no significant difference in test scores between the two types of meals, t(4) = 1.964; n.s. Type of test with degrees of freedom n.s. = “not significant” p<0.05 = “significant” Value of observed statistic Start summary with two means (based on DV) for two levels of the IV Finish with statistical summary t(4) = 1.96; ns Describe type of test (t-test versus Anova) with brief overview of results Or if it *were* significant: t(9) = 3.93; p < 0.05 39

Graphing your t-test results 40

Graphing your t-test results 41

Graphing your t-test results Chart Layout 42

Fill out titles 43

Where are we? Donald is a consultant and leads training sessions. As part of his training sessions, he provides the students with breakfast. He has noticed that when he provides a full breakfast people seem to learn better than when he provides just a small meal (donuts and muffins). So, he put his hunch to the test. He had two classes, both with three people enrolled. The one group was given a big meal and the other group was given only a small meal. He then compared their test performance at the end of the day. Please test with an alpha = .05 Big Meal 22 25 Small meal 19 23 21 Mean= 24 Mean= 21 We compared test scores for large and small meals. The mean test scores for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there appears to be no significant difference in test scores between the two types of meals, t(4) = 1.964; n.s. 44

What if we ran more subjects? Independent samples t-test Donald is a consultant and leads training sessions. As part of his training sessions, he provides the students with breakfast. He has noticed that when he provides a full breakfast people seem to learn better than when he provides just a small meal (donuts and muffins). So, he put his hunch to the test. This time he had two classes, both with nine people enrolled. The one group was given a big meal and the other group was given only a small meal. He then compared their test performance at the end of the day. Please test with an alpha = .05 Big Meal 22 25 Small meal 19 23 21 Mean= 21 Mean= 24 45

Notice: Additional participants don’t affect this part of the problem Independent samples t-test Notice: Additional participants don’t affect this part of the problem Step 1: Identify the research problem Did the size of the meal affect the test scores? Step 2: Describe the null and alternative hypotheses Ho: The size of the meal has no effect on test scores H1: The size of the meal does have an effect on test scores One tail or two tail test? 46

Notice: Two different ways to think about it Hypothesis testing Step 3: Decision rule α = .05 n1 = 9; n2 = 9 Degrees of freedom total (df total) = (n1 - 1) + (n2 – 1) = (9 - 1) + (9 – 1) = 16 Degrees of freedom total (df total) = (n total - 2) = 18 – 2 = 16 two tailed test Notice: Two different ways to think about it Critical t(16) = 2.12 47

two tail test α= .05 (df) = 16 Critical t(16) = 2.12 48

8 8 Step 4: Calculate observed t-score 18 2.25 Notice: s2 = 2.25 24 Mean= 21 Mean= 24 Big Meal Deviation From mean 2 -1 Small Meal Deviation From mean 2 -2 Squared deviation 4 1 Squared Deviation 4 Big Meal 22 25 Small meal 19 23 21 Σ = 18 Σ = 24 18 2.25 Notice: s2 = 2.25 1 8 1 Notice: Simple Average = 2.625 24 3.00 Notice: s2 = 3.0 2 2 8 49

Sp2 = 2.625 S21 = 2.25 S22 = 3.00 Step 4: Calculate observed t-score Mean= 21 Mean= 24 Big Meal 22 25 Small meal 19 23 21 Sp2 = 2.625 S21 = 2.25 S22 = 3.00 S2pooled = (n1 – 1) s12 + (n2 – 1) s22 n1 + n2 - 2 S2pooled = (9 – 1) (2.25) + (9 – 1) (3) 9 + 9 - 2 = 2.625 50

Sp2 = 2.625 S21 = 2.25 S22 = 3.00 Step 4: Calculate observed t-score Mean= 21 Mean= 24 Big Meal 22 25 Small meal 19 23 21 Sp2 = 2.625 S21 = 2.25 S22 = 3.00 = 24 – 21 0.7638 24 - 21 = 3.928 2.625 2.625 9 9 51

Step 5: Make decision whether or not to reject null hypothesis Summarizing your t-test results Step 5: Make decision whether or not to reject null hypothesis Observed t = 3.928 Critical t = 2.120 3.928 is farther out on the curve than 2.120 so, we do reject the null hypothesis t(16) = 3.928; p < 0.05 52

We compared test scores for large and small meals. The mean test Summarizing your t-test results Step 6: Conclusion We compared test scores for large and small meals. The mean test score for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there was a significant difference in test scores between the two types of meals t(16) = 3.928; p < 0.05 53

Let’s run more subjects using our excel! 54

Let’s run more subjects using our excel! Finding Means Finding Means 55

Let’s run more subjects using our excel! This is variance for each sample (Remember, variance is just standard deviation squared) Please note: “Pooled variance” is just like the average of the two sample variances, so notice that the average of 2.25 and 3 is 2.625 56

Let’s run more subjects using our excel! This is “n” for each sample (Remember, “n” is just number of observations for each sample) This is “n” for each sample (Remember, “n” is just number of observations for each sample) Remember, “degrees of freedom” is just (n-1) for each sample. So for sample 1: n-1 =9-1 = 8 And for sample 2: n-1=9-1 = 8 Then, df = 8+8=16 df = “degrees of freedom” 57

Let’s run more subjects using our excel! Finding degrees of freedom Finding degrees of freedom 58

Let’s run more subjects using our excel! Finding Observed t 59

Let’s run more subjects using our excel! Finding Critical t 60

Let’s run more subjects using our excel! Remember, if the “t Stat” is bigger than the “t Critical” then we “reject the null”, and conclude we have a significant effect Remember, if the “t Stat” is bigger than the “t Critical” then we “reject the null”, and conclude we have a significant effect 61

Let’s run more subjects using our excel! Finding p value (Is it less than .05?) 62

Let’s run more subjects using our excel! In this case, p = 0.0012 which is less than 0.05, so we “do reject the null” Remember, if the “p” is less than 0.05 then we “reject the null”, and conclude we have a significant effect 63

Let’s run more subjects using our excel! We compared test scores for large and small meals. The mean test score for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there was a significant difference in test scores between the two types of meals t(16) = 3.928; p < 0.05 Let’s run more subjects using our excel! 64

Thank you! See you next time!!