1. Reasoning in Psychology Using Statistics
2018

2. Many students found the lab exam too long/difficult to complete
Exam 3 extra credit opportunity (rather than a “curve”) for everybody (regardless of how you did on the exam)
May earn up to 15 points of extra-credit
This is a one time offer, the due date is: Monday April 16. It should be printed out and turned in at lecture that day.
late submissions will not be accepted
Exam 3 Extra-credit

3. Start working on your Final Projects soon (see link on syllabus page)
Due Wed, May 2 (uploaded to ReggieNet Assignment: Final Project)
Lab instructor assign a case in lab today (Wednesday)
Make sure to download:
Your case datafile
Expectations
Write in sentences and paragraphs. Don’t just copy and paste SPSS; also interpret the output. There is a “sample paper” provided.
Checklist
Need to run SPSS
During lab after finish lab exercise or Milner lab or DEG 17 (PRC)
PRC hours:
Final Projects

4. Decision tree Changing focus Looking for differences between groups:
ONE VARIABLE
Looking for relationships between TWO VARIABLES
Decision tree

5. Decision tree Changing focus
Looking for relationships between variables
(not looking for differences between groups)
Describing the strength of the relationship
Today’s topic: Pearson’s correlation
Quantitative variables
Two variables
Relationship between variables
Decision tree

6. Relationships between variables
Relationships between variables may be described with correlation procedures
Suppose that you notice that the more you study for an exam, the better your score typically is.
This suggests that there is a relationship between:
study time
test performance
115 mins
15 mins
Relationships between variables

7. Relationships between variables
Relationships between variables may be described with correlation procedures
To examine this relationship you should:
Make a Scatterplot Y X 1 2 3 4 5 6
Compute the Correlation Coefficient
Determine whether the correlation coefficient is statistically significant - hypothesis testing
New
Relationships between variables

8. Review & New -1.0 0.0 +1.0 Fail to Reject H0 Reject H0 rcritical
perfect negative corr.
r = 0.0
no relationship
r = 1.0
perfect positive corr.
Fail to Reject H0
Reject H0
rcritical
-1.0
0.0
+1.0
The farther from zero, the stronger the relationship
How strong a correlation to conclude it is beyond what expected by chance?
Review & New

9. Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Is there a statistically significant relationship between these variables (alpha = 0.05)? Y X 1 2 3 4 5 6 X Y A B C D E
Example

10. Review: Computing Pearson’s r
Pearson product-moment correlation
A numeric summary of the relationship
Step 1
Step 1: compute Sum of the Products (SP)
r =
degree to which X and Y vary together
degree to which X and Y vary separately
Step 3
Step 3: compute r
Step 2
Step 2: SSX & SSY
Review: Computing Pearson’s r

11. Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Is there a statistically significant relationship between these variables (alpha = 0.05)? X Y A
2.4
-2.6
1.4
-0.6
0.0
5.76
6.76
1.96
0.36
15.20
2.0
-2.0
0.0
4.0
0.0
16.0
14.0
4.8
5.2
2.8
0.0
1.2 B C D E
mean
3.6
4.0
SSY
SSX
Step 2 SP
Step 1
Example

12. Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Is there a statistically significant relationship between these variables (alpha = 0.05)?
Step 3 X Y A B C D E
15.20
SSX
16.0
SSY
14.0 SP
Example
Step 2
Step 1

13. Example Appears linear Positive relationship
Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Is there a statistically significant relationship between these variables (alpha = 0.05)? Y X 1 2 3 4 5 6
Appears linear
Positive relationship
Fairly strong relationship
.898 is far from 0, near +1 X Y A B C
Fairly strong, but stronger than you would expect by chance? D E
Example

14. Example Hypothesis testing
Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Is there a statistically significant relationship between these variables (alpha = 0.05)?
Hypothesis testing
Core logic of hypothesis testing
Considers the probability that the result of a study could have come about if no effect (in this case “no relationship”)
If this probability is low, then the scenario of no effect (relationship) is rejected Y X 1 2 3 4 5 6 X Y A B C
Fairly strong, but stronger than you would expect by chance? D E
Example

15. Example Step 1: State your hypotheses
Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Is there a statistically significant relationship between these variables (alpha = 0.05)?
Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data
Step 4: Compute your test statistics
Step 5: Make a decision about your null hypothesis X Y A B C D E
Example

16. Hypothesis testing with Pearson’s r
Step 1: State your hypotheses: as a research hypothesis and a null hypothesis about the populations
Null hypothesis (H0)
Research hypothesis (HA)
There are no correlation between the variables (they are independent)
ρ = 0
Generally, the variables correlated (they are not independent)
ρ ≠ 0
Note: symbol ρ (rho)
is actually correct,
but rarely used
Hypothesis testing with Pearson’s r

17. Hypothesis testing with Pearson’s r
Step 1: Hypotheses
Two -tailed
Hypothesize that variables are correlated (either direction)
H0:
ρ = 0
HA:
ρ ≠ 0
Hypothesis testing with Pearson’s r

18. Hypothesis testing with Pearson’s r
Step 1: Hypotheses
ρ ≥ 0
ρ < 0
H0:
HA:
Hypothesize that variables are:
One -tailed
Negatively correlated
Positively correlated
ρ < 0
ρ > 0
Two -tailed
Hypothesize that variables are correlated (either direction)
H0:
ρ = 0
HA:
ρ ≠ 0
Hypothesis testing with Pearson’s r

19. Suppose that you think that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). You decide to test whether there is any statistically significant relationship between these variables (alpha = 0.05).
Step 1
2-tailed X Y
There is no correlation between the study time and exam performance A B
ρ = 0
H0: C
There is a correlation between the study time and exam performance D
HA:
ρ ≠ 0 E
Example: New

20. Hypothesis testing with Pearson’s r
Step 1: Hypotheses
Step 2: Criterion for decision
Alpha (α) level as guide for when to reject or fail to reject the null hypothesis.
Based on probability of making type I error
Hypothesis testing with Pearson’s r

21. You decide to test whether there is any statistically significant relationship between these variables (alpha = 0.05).
2-tailed
ρ = 0
H0:
HA:
ρ ≠ 0 X Y
Step 2 A
α = 0.05 B C D E
Example: New

22. Hypothesis testing with Pearson’s r
Step 1: Hypotheses
Step 2: Criterion for decision
Steps 3 & 4: Sample & Test statistics
Descriptive statistics (Pearson’s r)
Degrees of freedom (df): df = n – 2
Used up one for each variable for calculating its mean
Note that n refers to number of pairs of scores, as in related-samples t-tests
Hypothesis testing with Pearson’s r

23. Example: New Steps 3 & 4 r = 0.898 df = n - 2 = 5 - 2 =3
You decide to test whether there is any statistically significant relationship between these variables (alpha = 0.05).
2-tailed
ρ = 0
H0:
HA:
ρ ≠ 0 X Y Y X 1 2 3 4 5 6
α = 0.05 A
Steps 3 & 4 B
r = 0.898 C
df = n - 2
= =3 D E
Example: New

24. Hypothesis testing with Pearson’s r
Step 1: Hypotheses
Step 2: Criterion for decision
Steps 3 & 4: Sample & Test statistics
Step 5: Compare observed and critical test values
Use the Pearson’s r table (based on t-test or r to z transformation)
Note: For very small df, need very large r for significance
Critical values of r (rcrit)
Hypothesis testing with Pearson’s r

25. Example: New df = n - 2 = 3 Step 5 rcrit = ±0.878
Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Is there a statistically significant relationship between these variables (alpha = 0.05)?
2-tailed
ρ = 0
H0:
HA:
ρ ≠ 0 X Y
df = n - 2
= 3
α = 0.05 Y X 1 2 3 4 5 6 A
Step 5
rcrit = ±0.878
From table B C D E
Example: New

26. Hypothesis testing with Pearson’s r
Step 1: Hypotheses
Step 2: Criterion for decision
Steps 3 & 4: Sample & Test statistics
Step 5: Compare observed and critical test values
& Make a decision about H0 & Conclusions
1-tailed case when H0: r > 0
-1.0
0.0
+1.0
rcritical
Fail to Reject H0
Reject H0
Hypothesis testing with Pearson’s r

27. r = 
H0:
HA:
r ≠ 
2-tailed
-1.0
0.0
+1.0
The observed correlation is farther away from zero than the rcritical so we reject H0
Suppose that you notice that the more you study for an exam (X= hours of study), the better your exam score typically is (Y = exam score). Is there a statistically significant relationship between these variables (alpha = 0.05)? X Y
df = n - 2
= 3
alpha = 0.05 A
rcrit = ±0.878 B Y X 1 2 3 4 5 6
Step 5 C
Reject H0
Conclude that the correlation is not equal to 0 D E
“There is a significant positive correlation between study time and exam performance”
Example: New

28. Generally, it is considered best to have at least 30 pairs of scores to conduct a Pearson’s r analysis
Minimum N = 30,
df = 28, rcrit = .30
Best Practice

29. Using Correlation in SPSS
SPSS: HGT.SAV Height by Weight, N = 40
Note that significance is expressed the same as previously
r (38) = .794, p < .001
What is p for 1-tailed test?
For df = 38, α = .05, 2-tailed, rcrit = .31
Using Correlation in SPSS

30. Wrap up In labs: Questions?
Hypothesis testing with correlation (by hand and with SPSS)
Questions?
Wrap up