2. Modern Languages 14131211109 87 6 54321 111098765 43 2 Row A Row B Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 212019181716 1514 13 12111098 212019181716 13 12111098 141312 table 7 6 54321 Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 321 21 1413 Projection Booth 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 212019181716 1514 13 12111098 7 6 5432 1 765 43 2 1 7 6 5432 1 765 43 2 1 7 6 54321 765 43 2 1 7 6 54321 765 43 2 1 7 6 54321 table Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 321 28 27 26252423 22 282726 2524 23 22 282726 2524 23 22 28 27 26252423 22 282726 2524 23 22 282726 2524 23 22 28 27 26252423 22 282726 2524 23 22 282726 2524 23 22 282726 2524 23 22 R/L handed broken desk Stage Lecturer’s desk Screen 1

3. MGMT 276: Statistical Inference in Management Spring 2015

5. Before our next exam (April 14 th ) Lind (10 – 12) Chapter 10: One sample Tests of Hypothesis Chapter 11: Two sample Tests of Hypothesis Chapter 12: Analysis of Variance Plous (2, 3, & 4) Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight Bias Chapter 4: Context Dependence Schedule of readings

6. Logic of hypothesis testing Steps for hypothesis testing Levels of significance (Levels of alpha) what does p < 0.05 mean? what does p < 0.01 mean? Hypothesis testing with t-scores (one-sample) Hypothesis testing with t-scores (two independent samples) Constructing brief, complete summary statements By the end of lecture today 3/26/15

7. On class website: Please print and complete homework worksheet #12 Hypothesis testing with z-tests and t-tests Homework due – Tuesday (March 31 st )

8. Exam 2 – Thanks for your patience and cooperation The grades will be posted by Tuesday It went really well!

9. Remember… In a negatively skewed distribution: mean < median < mode Score on Exam Note: Always “frequency” Note: Label and Numbers 80 = mode = tallest point 78 = median = middle score 77 = mean = balance point Median Mean Frequency Mode

10. 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 One or two tailed test? Balance between Type I versus Type II error Critical statistic (e.g. z or t or F or r) value?

11. Degrees of Freedom Degrees of Freedom ( d.f. ) is a parameter based on the sample size that is used to determine the value of the t statistic. Degrees of freedom tell how many observations are used to calculate s, less the number of intermediate estimates used in the calculation. We lose one degree of freedom for every parameter we estimate

12. Standard deviation and Variance For Sample and Population These would be helpful to know by heart – please memorize these formula Pop Quiz – Part 1

13. Standard deviation and Variance For Sample and Population Pop Quiz – Part 1 Part 2: When we move from a two-tailed test to a one-tailed test what happens to the critical z score (bigger or smaller?) - Draw a picture - What affect does this have on the hypothesis test (easier or harder to reject the null?)

14. Pop Quiz – Part 3 1. When do we use a t-test and when do we use a z-test? (Be sure to write out the formulae) 2. How many steps in hypothesis testing (What are they?) 3. What is our formula for degrees of freedom in one sample t-test? 4. We lose one degree of freedom for every ________________ 5. What are the three parts to the summary (below) The mean response time for following the sheriff’s new plan was 24 minutes, while the mean response time prior to the new plan was 30 minutes. A t-test was completed and there appears to be no significant difference in the response time following the implementation of the new plan t(9) = -1.71; n.s.

15. Part 2: When we move from a two-tailed test to a one-tailed test what happens to the critical z score (bigger or smaller?) - Draw a picture - What affect does this have on the hypothesis test (easier or harder to reject the null?) Standard deviation and Variance For Sample and Population Pop Quiz Critical value gets smaller Gets easier to reject the null

16. Pop Quiz Writing Assignment 1. When do we use a t-test and when do we use a z-test? (Be sure to write out the formulae) Population versus sample standard deviation Use the t-test when you don’t know the standard deviation of the population, and therefore have to estimate it using the standard deviation of the sample

17. Five steps to hypothesis testing Step 1: Identify the research problem (hypothesis) Describe the null and alternative hypotheses Step 2: Decision rule: find “critical z” score Alpha level? ( α =.05 or.01)? Step 3: Calculate observed z score Step 4: Compare “observed z” with “critical z” If observed z > critical z then reject null p < 0.05 and we have significant finding Step 5: Conclusion - tie findings back in to research problem One versus two-tailed test How is a t score similar to a z score? Same logic and same steps How is a t score different than a z score?

18. Writing Assignment Degrees of freedom =(df) = (n – 1) Degrees of freedom (df ) = (n 1 - 1) + (n 2 – 1) One sample t-test Two sample t-test First Sample Second Sample Parameter: Population standard deviation 3. What is our formula for degrees of freedom in one sample t-test? 4. We lose one degree of freedom for every parameter we estimate Sample standard deviation Use the word "parameter” when describing a whole population (not just a sample). Usually we don’t know about the whole population so we have guess by using what we know about our sample. A short-hand way to let the reader know it we are describing a population (a parameter) is to use a Greek letter – for example, σ for populations standard deviation, and an s for the sample. In a t-test we never know the population standard deviation (parameter σ) we have to estimate this one parameter (using “s”), so we lose one df our degree of freedom is n-1

19. Writing Assignment 5. What are the three parts to the summary (below) The mean response time for following the sheriff’s new plan was 24 minutes, while the mean response time prior to the new plan was 30 minutes. A t-test was completed and there appears to be no significant difference in the response time following the implementation of the new plan t(9) = -1.71; n.s. 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” 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

21. Hypothesis testing: one sample t-test Is the mean of my observed sample consistent with the known population mean or did it come from some other distribution? We are given the following problem: 800 students took a chemistry exam. Accidentally, 25 students got an additional ten minutes. Did this extra time make a significant difference in the scores? The average number correct by the large class was 74. The scores for the sample of 25 was 76, 72, 78, 80, 73 70, 81, 75, 79, 76 77, 79, 81, 74, 62 95, 81, 69, 84, 76 75, 77, 74, 72, 75 Please note: In this example we are comparing our sample mean with the population mean (One-sample t-test) L e t ’ s j u m p r i g h t i n a n d d o a t - t e s t

22. Hypothesis testing Step 1: Identify the research problem / hypothesis Describe the null and alternative hypotheses Ho:Ho: µ = 74 Did the extra time given to this sample of students affect their chemistry test scores = 74H1:H1: One tail or two tail test? µ

23. Hypothesis testing Step 2: Decision rule =.05 Degrees of freedom (df) = ( n - 1) = (25 - 1) = 24 two tail test n = 25

24. α =.05 (df) = 24 Critical t (24) = 2.064 two tail test

25. Hypothesis testing Step 3: Calculations µ = 74 = 76.44 76 72 78 80 73 70 81 75 79 76 77 79 81 74 62 95 81 69 84 76 75 77 74 72 75 76 – 76.44 72 – 76.44 78 – 76.44 80 – 76.44 73 – 76.44 70 – 76.44 81 – 76.44 75 – 76.44 79 – 76.44 76 – 76.44 77 – 76.44 79 – 76.44 81 – 76.44 74 – 76.44 62 – 76.44 95 – 76.44 81 – 76.44 69 – 76.44 84 – 76.44 76 – 76.44 75 – 76.44 77 – 76.44 74 – 76.44 72 – 76.44 75 – 76.44 = -0.44 = -4.44 = +1.56 = + 3.56 = -3.44 = -6.44 = +4.56 = -1.44 = +2.56 = -0.44 = +0.56 = +2.56 = +4.56 = -2.44 = -14.44 = +18.56 = +4.56 = -7.44 = +7.56 = -0.44 = -1.44 = +0.56 = -2.44 = -4.44 = -1.44 0.1936 19.7136 2.4336 12.6736 11.8336 41.4736 20.7936 2.0736 6.5536 0.1936 0.3136 6.5536 20.7936 5.9536 208.5136 344.4736 20.7936 55.3536 57.1536 0.1936 2.0736 0.3136 5.9536 19.7136 2.0736 x (x - x) (x - x) 2 868.16 24 = 6.01 N = 25 Σx = 1911 Σ(x- x) = 0 Σ(x- x) 2 = 868.16 ΣxΣx = N 1911 25 =

26. . Hypothesis testing Step 3: Calculations s = 6.01 = 76.44 - 74 1.20 = 2.03 µ = 74 = 76.44 N = 25 76.44 - 74 6.01 25 critical t Observed t (24) = 2.03

27. Hypothesis testing Step 4: Make decision whether or not to reject null hypothesis Step 6: Conclusion: The extra time did not have a significant effect on the scores 2.03 is not farther out on the curve than 2.064, so, we do not reject the null hypothesis Observed t (24) = 2.03 Critical t (24) = 2.064

28. Hypothesis testing: Did the extra time given to these 25 students affect their average test score? notice we are comparing a sample mean with a population mean: single sample t-test The mean score for those students who where given extra time was 76.44 percent correct, while the mean score for the rest of the class was only 74 percent correct. A t-test was completed and there appears to be no significant difference in the test scores for these two groups t(24) = 2.03; n.s. Start summary with two means (based on DV) for two levels of the IV Describe type of test (t-test versus z-test) with brief overview of results Finish with statistical summary t(24) = 2.03; ns Or if it had been different results that *were* significant: t(24) = -5.71; p < 0.05 Type of test with degrees of freedom n.s. = “not significant” p<0.05 = “significant” n.s. = “not significant” p<0.05 = “significant” Value of observed statistic

29. What if we had chosen a one-tail test? Step 1: Identify the research problem Describe the null and alternative hypotheses Step 2: Decision rule α =.05 Did the extra time given to this sample of students increase their chemistry test scores One tail or two tail test? Degrees of freedom (df) = (n - 1) Critical t (24) = = (25 - 1) = 24 ????? H1:H1: Ho:Ho: µ ≤ 74 µ > 74 Prediction is uni-directional α is all at one end so “critical t” changes

30. α =.05 (df) = 24 Critical t (24) = 1.711 one tail test

31. What if we had chosen a one-tail test? Step 1: Identify the research problem Describe the null and alternative hypotheses Step 2: Decision rule =.05 Did the extra time given to this sample of students increase their chemistry test scores One tail or two tail test? Degrees of freedom (df) = (n - 1) Critical t (24) = = (25 - 1) = 24 1.711 H1:H1: Ho:Ho: µ ≤ 74 µ > 74

32. Calculations (exactly same as two-tail test) Step 3: Calculations µ = 74 = 76.44 76 72 78 80 73 70 81 75 79 76 77 79 81 74 62 95 81 69 84 76 75 77 74 72 75 76 – 76.44 72 – 76.44 78 – 76.44 80 – 76.44 73 – 76.44 70 – 76.44 81 – 76.44 75 – 76.44 79 – 76.44 76 – 76.44 77 – 76.44 79 – 76.44 81 – 76.44 74 – 76.44 62 – 76.44 95 – 76.44 81 – 76.44 69 – 76.44 84 – 76.44 76 – 76.44 75 – 76.44 77 – 76.44 74 – 76.44 72 – 76.44 75 – 76.44 = -0.44 = -4.44 = +1.56 = + 3.56 = -3.44 = -6.44 = +4.56 = -1.44 = +2.56 = -0.44 = +0.56 = +2.56 = +4.56 = -2.44 = -14.44 = +18.56 = +4.56 = -7.44 = +7.56 = -0.44 = -1.44 = +0.56 = -2.44 = -4.44 = -1.44 0.1936 19.7136 2.4336 12.6736 11.8336 41.4736 20.7936 2.0736 6.5536 0.1936 0.3136 6.5536 20.7936 5.9536 208.5136 344.4736 20.7936 55.3536 57.1536 0.1936 2.0736 0.3136 5.9536 19.7136 2.0736 x (x - x) (x - x) 2 868.16 24 = 6.01 N = 25 Σx = 1911 Σ(x- x) = 0 Σ(x- x) 2 = 868.16 ΣxΣx = N 1911 25 = One-tailed test has no effect on calculations stage

33. . Calculations (exactly same as two-tail test) Step 3: Calculations s = 6.01 = 76.44 - 74 1.20 = 2.03 µ = 74 = 76.44 N = 25 76.44 - 74 6.01 25 critical t One-tailed test has no effect on calculations stage Observed t (24) = 2.03

34. Hypothesis testing Step 4: Make decision whether or not to reject null hypothesis Step 5: Conclusion: The extra time did have a significant effect on the scores 2.0 is farther out on the curve than 1.711, so, we do reject the null hypothesis Observed t (24) = 2.03 Critical t (24) = 1.711 Step 3: Calculations t (24) = 2.03 s = 6.01 µ = 74 = 76.44 N = 25

35. Hypothesis testing: Did the extra time given to these 25 students affect their average test score? notice we are comparing a sample mean with a population mean: single sample t-test The mean score for those students who where given extra time was 76.44 percent correct, while the mean score for the rest of the class was only 74 percent correct. A one-tailed t-test was completed and there appears to be significant difference in the test scores for these two groups t(24) = 2.03; p < 0.05. Start summary with two means (based on DV) for two levels of the IV Describe type of test (t-test versus z-test) with brief overview of results Finish with statistical summary t(24) = 2.03; ns Or if it had been different results that *were* significant: t(24) = -5.71; 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”

36. 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 One or two tailed test? Balance between Type I versus Type II error Critical statistic (e.g. z or t or F or r) value?

37. Hypothesis testing Step 1: Identify the research problem Step 2: Describe the null and alternative hypotheses H o : The response times are not changed Did the sheriff keep her promise to change response times from the previous average of 30 minutes? H 1 : The response times did change As the new chief of police, I am going to change response times for traffic accidents. Before I started the average response time was 30 minutes Is this a single sample or two sample test? Is it a z or a t test or an ANOVA? Is it a one-tail test or a two tail test?

38. Hypothesis testing Gather the data: We measured the time for police to respond to 10 accidents One or two tailed test? What is our sample size What is size of our degrees of freedom? What is our alpha What is our critical t value? n = 10 (df = 9) Alpha =.05 Decision rule: critical t = 1.83 Two-tailed test n = 10 df = 9 alpha = 0.05

39. Hypothesis testing Gather the data: We measured the time for police to respond to 10 accidents One or two tailed test? What is our sample size What is size of our degrees of freedom? What is our alpha What is our critical t value? Decision rule: critical t = 2.262 Two-tailed test Alpha of 0.05 Critical t (9) = 2.262

40. Step 3: Calculations: Average time for response before 30 minutes Average time for response after 24 minutes Observed t = - 2.71 Step 4: Make decision whether or not to reject null hypothesis -2.71 is farther out on the curve than 2.262 so, we do not reject the null hypothesis Observed t = - 2.71 Critical t = 2.262 Step 5: Conclusion: There appears to be a significant difference between the sheriff’s times and 30 minutes

41. Hypothesis testing: Did the sheriff keep her promise to reduce response times to less than 30 minutes? notice we are comparing a sample mean with a population mean: single sample t-test The mean response time for following the sheriff’s new plan was 24 minutes, while the mean response time prior to the new plan was 30 minutes. A t-test was completed and there appears to be a significant difference in the response time following the implementation of the new plan t(9) = -2.262; p < 0.05 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(9) = -5.71; p < 0.05 Or if it had been different results that *were not* significant: t(9) = -1.71; ns 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”

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