2. Practice The Neuroticism Measure = 23.32 S = 6.24 n = 54
How many people likely have a neuroticism score between 29 and 34?

3. Practice (29-23.32) /6.24 = .91 area = .3186 ( 34-23.32)/6.26 = 1.71
= .1378
.1378*54 = 7.44 or 7 people

4. Practice
On the next test I will give an A to the top 5 percent of this class.
The average test grade is with a SD of 6.98.
How many points on the test did you need to get to get an A?

5. Step 1: Sketch out question
.05

6. Step 2: Look in Table Z
Z score = 1.64
.05

7. Step 3: Find the X score that goes with the Z score
Must solve for X
X =  + (z)()
68.26 = (1.64)(6.98)

8. Step 3: Find the X score that goes with the Z score
Must solve for X
X =  + (z)()
68.26 = (1.64)(6.98)
Thus, a you need a score of to get an A

9. Practice
The prestigious Whatsamatta U will only take people scoring in the top 97% on the verbal section SAT (i.e., they reject the bottom 3%).
What is the lowest score you can get on the SAT and still get accepted?
Mean = 500; SD = 100

10. Step 1: Sketch out question
.03

11. Step 2: Look in Table C
Z score = -1.88
.03

12. Step 3: Find the X score that goes with the Z score
Must solve for X
X =  + (z)()
312 = (-1.88)(100)

13. Step 3: Find the X score that goes with the Z score
Must solve for X
X =  + (z)()
312 = (-1.88)(100)
Thus, you need a score of 312 on the verbal SAT to get into this school

14. 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)

15. 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

16. 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?

17. Basic logic of research

18. Start with two equivalent groups of subjects

19. Treat them alike except for one thing

20. See if both groups are different at the end

21. Or – Single Group

22. Do something

23. Measure DV

24. Compare Group to Population
Population Happiness Score

25. The Theory of Hypothesis Testing
Data are ambiguous
Is a difference due to chance?
Sampling error

26. Population
You are interested in the average self-esteem in a population of 40 people
Self-esteem test scores range from 1 to 10.

27. Population Scores
1,1,1,1
2,2,2,2
3,3,3,3
4,4,4,4
5,5,5,5
6,6,6,6
7,7,7,7
8,8,8,8
9,9,9,9
10,10,10,10

28. Histogram

29. What is the average self-esteem score of this population?
Population mean = 5.5
Population SD = 2.87
What if you wanted to estimate this population mean from a sample?

30. What if
Randomly select 5 people and find the average score

31. Group Activity
Why isn’t the average score the same as the population score?
When you use a sample there is always some degree of uncertainty!
We can measure this uncertainty with a sampling distribution of the mean

32. EXCEL

33. INTERNET EXAMPLE

34. Sampling Distribution of the Mean
Notice: The sampling distribution is centered around the population mean!
Notice: The sampling distribution of the mean looks like a normal curve!
This is true even though the distribution of scores was NOT a normal distribution

35. Central Limit Theorem
For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution a N (sample size) get larger. Furthermore, the sampling distribution of the mean will have a mean equal to  and a standard deviation equal to / N

36. Sampling Distribution
Tells you the probability of a particular sample mean occurring for a specific population

37. Sampling Distribution
You are interested in if your new Self-esteem training course worked.
The 5 people in your course had a mean self-esteem score of 5.5

38. Sampling Distribution
Did it work?
How many times would we expect a sample mean to be 5.5 or greater?
Theoretical vs. empirical
5,000 random samples yielded 2,501 with means of 5.5 or greater
Thus p = of this happening

39. Sampling Distribution
5.5
P = P =.5002
2, ,501

40. Sampling Distribution
You are interested in if your new Self-esteem training course worked.
The 5 people in your course had a mean self-esteem score of 5.8

41. Sampling Distribution
Did it work?
How many times would we expect a sample mean to be 5.8 or greater?
5,000 random samples yielded 2,050 with means of 5.8 or greater
Thus p = .41 of this happening

42. Sampling Distribution
5.8
P = P =.41
2, ,300

43. Sampling Distribution
The 5 people in your course had a mean self-esteem score of 9.8.
Did it work?
5,000 random samples yielded 4 with means of 9.8 or greater
Thus p = of this happening

44. Sampling Distribution
9.8
P = P =.0008 4,

45. Logic 1) Research hypothesis 2) Collect data
Training increased self-esteem
The sample mean is greater than general population mean
2) Collect data
3) Set up the null hypothesis H0
Training did not increase self-esteem
The sample is no different than general population mean

46. Logic
4) Obtain a sampling distribution of the mean under the assumption that H0 is true
5) Given the distribution obtain a probability of a mean at least as large as our actual sample mean
6) Make a decision
Either reject H0 or fail to reject H0

47. Hypothesis Test – Single Subject
You think your IQ is “freakishly” high that you do not come from the population of normal IQ adults.
Population IQ = 100 ; SD = 15
Your IQ = 125

48. Step 1 and 3
H1: 125 > μ
Ho: 125 < or = μ

49. Step 4: Appendix Z shows distribution of Z scores under null
-3 -2 -1  1 2  3 

50. Step 5: Obtain probability
125
-3 -2 -1  1 2  3 

51. Step 5: Obtain probability
( ) / 15 = 1.66
125
-3 -2 -1  1 2  3 

52. Step 5: Obtain probability
Z = 1.66
125
.0485
-3 -2 -1  1 2  3 

53. Step 6: Decision
Probability that this score is from the same population as normal IQ adults is .0485
In psychology
Most common cut-off point is p < .05
Thus, your IQ is significantly HIGHER than the average IQ

54. One vs. Two Tailed Tests
Previously wanted to see if your IQ was HIGHER than population mean
Called a “one-tailed” test
Only looking at one side of the distribution
What if we want to simply determine if it is different?

55. One-Tailed -3 -2 -1  1 2  3  H1: IQ > μ Ho: IQ < or = μ
p = .05 μ
-3 -2 -1  1 2  3 
Did you score HIGHER than population mean?
Want to see if score falls in top .05

56. Two-Tailed -3 -2 -1  1 2  3  H1: IQ = μ Ho: IQ = μ p = .05
-3 -2 -1  1 2  3 
Did you score DIFFERNTLY than population mean?

57. Two-Tailed -3 -2 -1  1 2  3  H1: IQ = μ Ho: IQ = μ p = .05
-3 -2 -1  1 2  3 
Did you score DIFFERNTLY than population mean?
PROBLEM: Above you have a p = .10, but you want to test at a p = .05

58. Two-Tailed -3 -2 -1  1 2  3  H1: IQ = μ Ho: IQ = μ p = .025
-3 -2 -1  1 2  3 
Did you score DIFFERNTLY than population mean?

59. Step 6: Decision
Probability that this score is from the same population as normal IQ adults is .0485
In psychology
Most common cut-off point is p < .05
Note that on the 2-tailed test the point of significance is .025 (not .05)
Thus, your IQ is not significantly DIFFERENT than the average IQ

61. Problems Problems with Null hypothesis testing Logic is backwards:
Most think we are testing the probability of the hypothesis given the data
Really testing the probability of the data given the null hypothesis!

62. Practice
A recently admitted class of graduate students at a large university has a mean GRE verbal score of 650 with a SD of 50. One student, whose mom is on the board of trustees, has a GRE score of Do you think the school was showing favoritism?
Why is there such a small SD?
Why might (or might not) the GRE scores in this sample be normally distributed?

63. 4.7
Z = ( ) / 50 = -3.2
p = (490 or lower)

64. 4.8
Because students are being selected with high GREs (restricted range)

65. 4.9
Would not be normally distributed
Negatively skewed

66. Practice
Last nights NHL game resulted in a score of 26 – 13. You would probably guess that I misread the paper. In effect you have just tested and rejected a null hypothesis.
1) What is the null hypothesis
2) Outline the hypothesis testing precede you just applied.

67. 4.1
a) Null = last nights game was an NHL game (i.e., the scores come from the population of all NHL scores)
B) Would expect that a team would score between 0 – 6 points (null hypothesis).
Because the actual scores are a lot different we would reject the null.