2. Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Spring 2015 Room 150 Harvill Building 8:00 - 8:50 Mondays, Wednesdays & Fridays.

4. Guess how much this bull weighs? http://www.npr.org/blogs/parallels/2014/04/02/297839429/-so-you-think-youre-smarter-than-a-cia-agent 1587.33 How old is the oldest person you know? How tall is the statue of liberty?

5. Schedule of readings Before next exam (April 10 th ) Please read chapters 7 – 11 in Ha & Ha Please read Chapters 2, 3, and 4 in Plous Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence

6. By the end of lecture today 3/30/15 Use this as your study guide Logic of hypothesis testing Steps for hypothesis testing Levels of significance (Levels of alpha) Hypothesis testing with t-scores (two independent samples) Constructing brief, complete summary statements Using Excel for completing t-tests

7. Homework due Assignment 16 Two-sample t-tests Due: Wednesday, April 1 st

8. Labs continue this week Project 2

9. Presentation Skills

13. mean + z σ = 30 ± (1.96)(2) mean + z σ = 30 ± (2.58)(2) 26.08 < µ < 33.92 24.84 < µ < 35.16 95% 99%

14. Melvin Mark Melvin Difference not due sample size because both samples same size Difference not due population variability because same population Yes! Difference is due to sloppiness and random error in Melvin’s sample Melvin

15. Ho: µ = 5 Ha: µ ≠ 5 Bags of potatoes from that plant are not different from other plants Bags of potatoes from that plant are different from other plants Two tailed test (α =.05) 1.96 6 – 5.25 = 4.0 1 16 √ =.25 4.0 1.96 -1.96 1 4 = z- score : because we know the population standard deviation

16. Yes These three will always match Probability of Type I error is always equal to alpha.05 Because the observed z (4.0 ) is bigger than critical z (1.96) 1.64 No Because observed z (4.0) is still bigger than critical z (1.64) 2.58 there is a difference No Because observed z (4.0) is still bigger than critical z(2.58) there is no difference there is not there is 1.96 2.58

17. Two tailed test (α =.05) Critical t(15) = 2.131 89 - 85 6 16 √ 2.667 t- score : because we don’t know the population standard deviation n – 1 =16 – 1 = 15 2.13 -2.13

18. α =.05 two tail test (df) = 15 Critical t (15) = 2.131

19. α =.05 (df) = 15 Critical t (15) = 2.131 two tail test

20. Yes These three will always match Probability of Type I error is always equal to alpha.05 Because the observed z (2.67) is bigger than critical z (2.13) 1.753 No Because observed t (2.67) is still bigger than critical t (1.753) 2.947 consultant did improve morale Yes Because observed t (2.67) is not bigger than critical t(2.947) consultant did not improve morale she did not she did 2.131 2.947 No These three will always match

21. The average weight of bags of potatoes from this particular plant is 6 pounds, while the average weight for population is 5 pounds. A z-test was completed and this difference was found to be statistically significant. We should fix the plant. (z = 4.0; p<0.05) Start summary with two means (based on DV) for two levels of the IV Describe type of test (z-test versus t-test) with brief overview of results Finish with statistical summary z = 4.0; p < 0.05 Or if it *were not* significant: z = 1.2 ; n.s. Value of observed statistic n.s. = “not significant” p<0.05 = “significant”

22. The average job-satisfaction score was 89 for the employees who went On the retreat, while the average score for population is 85. A t-test was completed and this difference was found to be statistically significant. We should hire the consultant. (t(15) = 2.67; p<0.05) Start summary with two means (based on DV) for two levels of the IV Describe type of test (z-test versus t-test) with brief overview of results Finish with statistical summary t(15) = 2.67; p < 0.05 Or if it *were not* significant: t(15) = 1.07; n.s. df Value of observed statistic n.s. = “not significant” p<0.05 = “significant”

23. 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)? 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 Critical statistic (e.g. z or t) value? How is a t score same as a z score? Population versus sample standard deviation How is a t score different than a z score?

24. Comparing z score distributions with t-score distributions 1)We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample Critical t (just like critical z) separates common from rare scores Critical t used to define both common scores “confidence interval” and rare scores “region of rejection”

25. Comparing z score distributions with t-score distributions 2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n) Please notice: as sample sizes get smaller, the tails get thicker. As sample sizes get bigger tails get thinner and look more like the z-distribution 1)We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample

26. .. A note on z scores, and t score: Difference between means Numerator is always distance between means (how far away the distributions are) Denominator is always measure of variability (how wide or much overlap there is between distributions) Variability of curve(s)

27. 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)? 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 Critical statistic (e.g. z or t) value? How is a single sample t-test different than two sample t-test? Single sample standard deviation versus average standard deviation for two samples How is a single sample t-test most similar to the two sample t-test? Single sample has one “n” while two samples will have an “n” for each sample

28. 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= 24 Mean= 21 t = x 1 – x 2 variability t = 24 – 21 variability Got to figure this part out: We want to average from 2 samples - Call it “pooled” Are the two means significantly different from each other, or is the difference just due to chance?

29. Hypothesis testing Step 1: Identify the research problem Step 2: Describe the null and alternative hypotheses Did the size of the meal affect the learning / test scores? Step 3: Decision rule α =.05 Two tailed test Degrees of freedom total (df total ) = (n 1 - 1) + (n 2 – 1) = (3 - 1) + (3 – 1) = 4 n 1 = 3; n 2 = 3 Critical t (4) = 2.776 Step 4: Calculate observed t score Notice: Two different ways to think about it

30. α =.05 (df) = 4 Critical t (4) = 2.776 two tail test

31. 3 4 Mean= 24 Squared Deviation 4 0 Σ = 8 Big Meal 22 25 Small meal 19 23 21 Big Meal Deviation From mean -2 1 Squared deviation 4 1 Mean= 21 Small Meal Deviation From mean -2 2 0 Σ = 6 = 3.5 S 2 pooled = (n 1 – 1) s 1 2 + (n 2 – 1) s 2 2 n 1 + n 2 - 2 S 2 pooled = (3 – 1) (3) + (3 – 1) (4) 3 1 + 3 2 - 2 6 2 1 8 2 1 2 2 Notice: s 2 = 3.0 Notice: s 2 = 4.0 Notice: Simple Average = 3.5

32. Mean= 24 Squared Deviation 4 0 Σ = 8 Participant 1 2 3 Big Meal 22 25 Small meal 19 23 21 Big Meal Deviation From mean -2 1 Squared deviation 4 1 Mean= 21 Small Meal Deviation From mean -2 2 0 Σ = 6 = 24 – 21 1.5275 = 1.964 S 2 p = 3.5 24 - 21 3.5 33 Observed t 1.964 is not larger than 2.776 so, we do not reject the null hypothesis t(4) = 1.964; n.s. Observed t = 1.964 Critical t = 2.776 Conclusion: There appears to be no difference between the groups

33. How to report the findings for a t-test One paragraph summary of this study. Describe the IV & DV. Present the two means, which type of test was conducted, and the statistical results. Observed t = 1.964 df = 4 Mean of big meal was 24 Mean of small meal was 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. Start summary with two means (based on DV) for two levels of the IV Describe type of test (t-test versus anova) with brief overview of results Finish with statistical summary t(4) = 1.96; ns Or if it *were* significant: t(9) = 3.93; p < 0.05 Type of test with degrees of freedom Value of observed statistic n.s. = “not significant” p<0.05 = “significant” n.s. = “not significant” p<0.05 = “significant”

34. 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. Start summary with two means (based on DV) for two levels of the IV Describe type of test (t-test versus anova) with brief overview of results Finish with statistical summary t(4) = 1.96; ns Or if it *were* significant: t(9) = 3.93; p < 0.05 Type of test with degrees of freedom Value of observed statistic n.s. = “not significant” p<0.05 = “significant”