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 Null hypothesis H1: 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 Reject H0:  sample = 4 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 Do not reject H0:  sample = 4 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

29. 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”

30. 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

31. Hypothesis Testing Basic Logic
5) On the basis of that probability 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

33. 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

34. The Steps
Try to always follow these steps!

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

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

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

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

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

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

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

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

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

44. 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

45. 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.

46. 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

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

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

49. Step 3: Draw Critical Region
tcrit =
tcrit = 2.042

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

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

52. Step 4: Calculate t observed
tobs = (X - ) / Sx
.27 = 1.52 / 31

53. Step 4: Calculate t observed
tobs = (X - ) / Sx
.44 = ( ) / .27

54. Step 5: See if tobs falls in the critical region
tcrit =
tcrit = 2.042

55. Step 5: See if tobs falls in the critical region
tcrit =
tcrit = 2.042
tobs = .44

56. 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

57. Step 7: Put answer into words
We fail to reject H0
The average agreeableness score of Villanova students is not statistically different ( = .05) than the average agreeableness score of the population.