Is this quarter fair?

2. Is this quarter fair?

3. Is this quarter fair? • How could you determine this? • You assume that flipping the coin a large number of times would result in heads half the time (i.e., it has a .50 probability)

4. Is this quarter fair? • Say you flip it 100 times • 52 times it is a head • Not exactly 50, but its close • probably due to random error

5. Is this quarter fair? • What if you got 65 heads? • 70? • 95? • At what point is the discrepancy from the expected becoming too great to attribute to chance?

6. Example • You give 100 random students a questionnaire designed to measure attitudes toward living in dormitories • Scores range from 1 to 7 • (1 = unfavorable; 4 = neutral; 7 = favorable) • You wonder if the mean score of the population is different then the population mean at Haverford (which is 4)

7. Hypothesis • Alternative hypothesis • H1: sample = 4 • In other words, the population mean will be different than 4

8. Hypothesis • Alternative hypothesis • H1: sample = 4 • Null hypothesis • H0: sample = 4 • In other words, the population mean will not be different than 4

9. Results • N = 100 • X = 4.51 • s = 1.94 • Notice, your sample mean is consistent with H1, but you must determine if this difference is simply due to chance

10. Results • N = 100 • X = 4.51 • s = 1.94 • To determine if this difference is due to chance you must calculate an observed t value

11. Observed t-value tobs = (X - ) / Sx

12. Observed t-value tobs = (X - ) / Sx • This will test if the null hypothesis H0:  sample = 4 is true • The bigger the tobs the more likely that H1:  sample = 4 is true

13. Observed t-value tobs = (X - ) / Sx Sx = S / N

14. Observed t-value tobs = (X - ) / .194 .194 = 1.94/ 100

15. Observed t-value tobs = (4.51 – 4.0) / .194

16. Observed t-value 2.63 = (4.51 – 4.0) / .194

17. t distribution

18. t distribution tobs = 2.63

19. t distribution tobs = 2.63 Next, must determine if this t value happened due to chance or if represent a real difference in means. Usually, we want to be 95% certain.

20. t critical • To find out how big the tobs must be to be significantly different than 0 you find a tcrit value. • Calculate df = N - 1 • Table D • First Column are df • Look at an alpha of .05 with two-tails

21. t distribution tobs = 2.63

22. t distribution tcrit = -1.98 tcrit = 1.98 tobs = 2.63

23. t distribution tcrit = -1.98 tcrit = 1.98 tobs = 2.63

24. t distribution tcrit = -1.98 tcrit = 1.98 tobs = 2.63 If tobs fall in critical area reject the null hypothesis Reject H0:  sample = 4

25. t distribution tcrit = -1.98 tcrit = 1.98 tobs = 2.63 If tobs does not fall in critical area do not reject the null hypothesis Do not reject H0:  sample = 4

26. Decision • Since tobs falls in the critical region we reject Ho and accept H1 • It is statistically significant, the average favorability of Villanova dorms is significantly different than the favorability of Haverford dorms. • p < .05

27. p < .05 • We usually test for significance at the “.05 level” • This means that the results we got in the previous example would only happen 5 times out of 100 if the true population mean was really 4

30. Hypothesis Testing

31. Hypothesis Testing • Basic Logic • 1) Want to test a hypothesis (called the research or alternative hypothesis). • “Second born children are smarter than everyone else (Mean IQ of everyone else = 100”) • 2) Set up the null hypothesis that your sample was drawn from the general population • “Your sample of second born children come from a population with a mean of 100”

32. Hypothesis Testing • Basic Logic • 3) Collect a random sample • You collect a sample of second born children and find their mean IQ is 145 • 4) Calculate the probability of your sample mean occurring given the null hypothesis • What is the probability of getting a sample mean of 145 if they were from a population mean of 100

33. Hypothesis Testing • Basic Logic • 5) On the basis of that probably you make a decision to either reject of fail to reject the null hypothesis. • If it is very unlikely (p < .05) to get a mean of 145 if the population mean was 100 you would reject the null • Second born children are SIGNIFICANTLY smarter than the general population

34. Example • You wonder if the average IQ score of Villanova students is significantly different (at alpha = .05)than the average IQ of the population (which is 100). To determine this you collect a sample of 54 students. • N = 54 • X = 130 • s = 18.4

35. The Steps • Try to always follow these steps!

36. Step 1: Write out Hypotheses • Alternative hypothesis • H1: sample = 100 • Null hypothesis • H0: sample = 100

37. Step 2: Calculate the Critical t • N = 54 • df = 53 •  = .05 • tcrit = 2.0

38. Step 3: Draw Critical Region tcrit = -2.00 tcrit = 2.00

39. Step 4: Calculate t observed tobs = (X - ) / Sx

40. Step 4: Calculate t observed tobs = (X - ) / Sx Sx = S / N

41. Step 4: Calculate t observed tobs = (X - ) / Sx 2.5 = 18.4 / 54

42. Step 4: Calculate t observed tobs = (X - ) / Sx 12 = (130 - 100) / 2.5 2.5 = 18.4 / 54

43. Step 5: See if tobs falls in the critical region tcrit = -2.00 tcrit = 2.00

44. Step 5: See if tobs falls in the critical region tcrit = -2.00 tcrit = 2.00 tobs = 12

45. Step 6: Decision • If tobs falls in the critical region: • Reject H0, and accept H1 • If tobs does not fall in the critical region: • Fail to reject H0

46. Step 7: Put answer into words • We reject H0 and accept H1. • The average IQ of Villanova students statistically different ( = .05) than the average IQ of the population.

47. Practice • You wonder if the average agreeableness score of Villanova students is significantly different (at alpha = .05) than the average agreeableness of the population (which is 3.8). You collect data from 31 people. • N = 31 • X = 3.92 • s = 1.52

49. Step 1: Write out Hypotheses • Alternative hypothesis • H1: sample = 3.8 • Null hypothesis • H0: sample = 3.8

50. Step 2: Calculate the Critical t • N = 31 • df = 30 •  = .05 • tcrit = 2.042