1. Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2018 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays.
Welcome
3/19/18

2. Lecturer’s desk Projection Booth Screen Screen Harvill 150 renumbered
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Projection Booth
Left handed desk

4. Before next exam (April 6th)
Schedule of readings
Before next exam (April 6th)
Please read chapters in OpenStax textbook
Please read Chapters 2, 3, and 4 in Plous
Chapter 2: Cognitive Dissonance
Chapter 3: Memory and Hindsight Bias
Chapter 4: Context Dependence

5. Labs continue this week Project 3
Lab sessions
Labs continue this week Project 3

6. Comparing more than two means
One-way Analysis of Variance (ANOVA)
Prep
Project 3

8. 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)?
Critical statistic (e.g. z or t) value?
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
Review

9. Hypothesis testing with t-tests
The result is “statistically significant” if:
the observed statistic is larger than the critical statistic
observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x2) to be big!!
the p value is less than 0.05 (which is our alpha)
p < If we want to reject the null, we want our “p” to be small!!
we reject the null hypothesis
then we have support for our alternative hypothesis
Review

10. Independent samples t-test
Are the two means significantly different from each other, or is the difference just due to chance?
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= 21
Mean= 24
Got to figure this part out: We want to average from 2 samples - Call it “pooled”
x1 – x2
t =
24 – 21
variability
t =
variability
Review 10

11. Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25
Small meal 19 23 21 11

12. Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25
Small meal 19 23 21 12

13. Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25
Small meal 19 23 21
If checked you’ll want to include the labels in your variable range
If checked, you’ll want to include the labels in your variable range
If checked you’ll want to include the labels in your variable range 13

14. Complete a t-test
Finding Means
Finding Means 14

15. Complete a t-test This is variance for each sample
(Remember, variance is just standard deviation squared)
Please note:
“Pooled variance” is just like the average of the two sample variances, so notice that the average of 3 and 4 is 3.5 15

16. Complete a t-test This is “n” for each sample
(Remember, “n” is just number of observations for each sample)
This is “n” for each sample
(Remember, “n” is just number of observations for each sample)
Remember, “degrees of freedom” is just (n-1) for each sample.
So for sample 1: n-1 =3-1 = 2
And for sample 2: n-1=2-1 = 2
Then, df = 2+2=4
df = “degrees of freedom” 16

17. Finding degrees of freedom
Complete a t-test
Finding degrees of freedom 17

18. Complete a t-test
Finding Observed t 18

19. Complete a t-test
Finding Critical t 19

20. Finding Critical t 20

21. Finding p value (Is it less than .05?)
Complete a t-test
Finding p value (Is it less than .05?) 21

22. Step 4: Make decision whether or not to reject null hypothesis
Complete a t-test
Step 4: Make decision whether or not to reject null hypothesis
Reject when:
observed stat > critical stat
is not bigger than 2.776
“p” is less than 0.05 (or whatever alpha is)
p = is not less than 0.05
Step 5: Conclusion - tie findings back in to research problem
There was no significant difference, there is no
evidence that size of meal affected test scores 22

23. Type of test with degrees of freedom Value of observed statistic
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.
Type of test with degrees of freedom
n.s. = “not significant”
p<0.05 = “significant”
Value of observed statistic
Start summary with two means (based on DV) for two levels of the IV
Finish with statistical summary t(4) = 1.96; ns
Describe type of test (t-test versus Anova) with brief overview of results
Or if it *were* significant:
t(9) = 3.93; p < 0.05 23

24. Graphing your
t-test results 24

25. Graphing your
t-test results 25

26. Graphing your
t-test results
Chart Layout 26

27. Fill out titles 27

28. Where are we?
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
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. 28

29. A note on z scores, and t score: . .
A note on z scores, and t score:
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)
Difference between means
Difference between means
Variability of curve(s) (within group variability)
Variability of curve(s)

30. Effect size is considered relative to variability of distributions.
Effect size is considered relative to variability of distributions
1. Larger variance harder to find significant difference
Treatment
Effect x
Treatment
Effect
2. Smaller variance easier to find significant difference x

31. Effect size is considered relative to variability of distributions.
Effect size is considered relative to variability of distributions
Treatment
Effect x
Difference between means
Treatment
Effect x
Variability of curve(s) (within group variability)

32. A note on variability versus effect size Difference between means.
A note on variability
versus effect size
Difference between means
Difference between means
Variability of curve(s)
Variability of curve(s) (within group variability)

33. A note on variability versus effect size Difference between means.
A note on variability
versus effect size
Difference between means
Difference between means .
Variability of curve(s)
Variability of curve(s) (within group variability)

34. Hypothesis testing: A review.
Difference between means
Hypothesis testing: A review
Variability of curve(s)
If the observed stat is more extreme than the critical stat in the distribution (curve):
then it is so rare, (taking into account the variability) we conclude it must be from some other distribution
decision considers effect size and variability
then we reject the null hypothesis – we have a significant result
then we have support for our alternative hypothesis
p < (p < α)
If the observed stat is NOT more extreme than the critical stat in the distribution (curve):
then we know it is a common score (either because the effect size is too small or because the variability is to big) and is likely to be part of this null distribution,
we conclude it must be from this distribution
decision considers effect size and variability – could be overly variable
then we do not reject the null hypothesis
then we do not have support for our alternative hypothesis
p not less than (p not less than α) p is n.s.
Difference between means
critical statistic
critical statistic
Variability of curve(s)
(within group variability)
Variability of curve(s)
Review

35. What if we ran more subjects?
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. This time he had two classes, both with nine 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= 21
Mean= 24 35

36. Notice: Additional participants don’t affect this part of the problem
Independent samples t-test
Notice: Additional participants don’t affect this part of the problem
Step 1: Identify the research problem
Did the size of the meal affect the test scores?
Step 2: Describe the null and alternative hypotheses
Ho: The size of the meal has no effect on
test scores
H1: The size of the meal does have an effect on
test scores
One tail or two tail test? 36

37. Notice: Two different ways to think about it
Hypothesis testing
Step 3: Decision rule
α = .05
n1 = 9; n2 = 9
Degrees of freedom total (df total) = (n1 - 1) + (n2 – 1)
= (9 - 1) + (9 – 1) = 16
Degrees of freedom total (df total) = (n total - 2) = 18 – 2 = 16
two tailed test
Notice: Two different ways to think about it
Critical t(16)
= 2.12 37

38. two tail test
α= .05
(df) = 16
Critical t(16)
= 2.12 38

39. 8 8 Step 4: Calculate observed t-score 18 2.25 Notice: s2 = 2.25 24
Mean= 21
Mean= 24
Big Meal
Deviation
From mean 2 -1
Small Meal
Deviation
From mean 2 -2
Squared
deviation 4 1
Squared Deviation 4
Big Meal 22 25
Small meal 19 23 21
Σ = 18
Σ = 24 18
2.25
Notice: s2 = 2.25 1 8 1
Notice:
Simple Average = 2.625 24
3.00
Notice: s2 = 3.0 2 2 8 39

40. Sp2 = 2.625 S21 = 2.25 S22 = 3.00 Step 4: Calculate observed t-score
Mean= 21
Mean= 24
Big Meal 22 25
Small meal 19 23 21
Sp2 = 2.625
S21 = 2.25
S22 = 3.00
S2pooled =
(n1 – 1) s12 + (n2 – 1) s22
n1 + n2 - 2
S2pooled =
(9 – 1) (2.25) + (9 – 1) (3)
= 2.625 40

41. Sp2 = 2.625 S21 = 2.25 S22 = 3.00 Step 4: Calculate observed t-score
Mean= 21
Mean= 24
Big Meal 22 25
Small meal 19 23 21
Sp2 = 2.625
S21 = 2.25
S22 = 3.00 =
24 – 21
0.7638 =
2.625
2.625 9 9 41

42. Step 5: Make decision whether or not to reject null hypothesis
Summarizing your
t-test results
Step 5: Make decision whether or not to reject null hypothesis
Observed t =
Critical t =
3.928 is farther out on the curve than 2.120
so, we do reject the null hypothesis
t(16) = 3.928; p < 0.05 42

43. We compared test scores for large and small meals. The mean test
Summarizing your
t-test results
Step 6: Conclusion
We compared test scores for large and small meals. The mean test
score for the big meal was 24, and was 21 for the small meal.
A t-test was calculated and there was a significant difference in
test scores between the two types of meals t(16) = 3.928; p < 0.05 43

44. Let’s run more subjects using our excel!44

45. Let’s run more subjects using our excel!
Finding Means
Finding Means 45

46. Let’s run more subjects using our excel!
This is variance for each sample
(Remember, variance is just standard deviation squared)
Please note:
“Pooled variance” is just like the average of the two sample variances, so notice that the average of 2.25 and 3 is 2.625 46

47. Let’s run more subjects using our excel!
This is “n” for each sample
(Remember, “n” is just number of observations for each sample)
This is “n” for each sample
(Remember, “n” is just number of observations for each sample)
Remember, “degrees of freedom” is just (n-1) for each sample.
So for sample 1: n-1 =9-1 = 8
And for sample 2: n-1=9-1 = 8
Then, df = 8+8=16
df = “degrees of freedom” 47

48. Let’s run more subjects using our excel!
Finding degrees of freedom
Finding degrees of freedom 48

49. Let’s run more subjects using our excel!
Finding Observed t 49

50. Let’s run more subjects using our excel!
Finding Critical t 50

51. Let’s run more subjects using our excel!
Remember, if the “t Stat” is bigger than the “t Critical” then we “reject the null”, and conclude we have a significant effect
Remember, if the “t Stat” is bigger than the “t Critical” then we “reject the null”, and conclude we have a significant effect 51

52. Let’s run more subjects using our excel!
Finding p value (Is it less than .05?) 52

53. Let’s run more subjects using our excel!
In this case, p = which is less than 0.05, so we
“do reject the null”
Remember, if the “p” is less than 0.05 then we “reject the null”, and conclude we have a significant effect 53

54. Let’s run more subjects using our excel!
We compared test scores for large and small meals. The mean test
score for the big meal was 24, and was 21 for the small meal.
A t-test was calculated and there was a significant difference in
test scores between the two types of meals t(16) = 3.928; p < 0.05
Let’s run more subjects using our excel! 54

55. What happened? We ran more subjects: Increased n
So, we decreased variability
Easier to find effect significant
even though effect size didn’t change
This is the sample size
This is the sample size
Small sample
Big sample 55

56. What happened? We ran more subjects: Increased n
So, we decreased variability
Easier to find effect significant
even though effect size didn’t change
This is variance for each sample
(Remember, variance is just standard deviation squared)
This is variance for each sample
(Remember, variance is just standard deviation squared)
Small sample
Big sample 56

57. Thank you!
See you next time!!