1. Reasoning in Psychology Using Statistics
2015

2. Start working on your Final Projects soon (see link on syllabus page)
Due Wed, April 29, in labs
Lab instructor assign a case in lab on Monday
Make sure to download:
1) Your case datafile
2) Directions
Write in sentences and paragraphs. Don’t just copy and paste SPSS; also interpret the output.
3) Checklist (and attach to what you turn in)
Need to run SPSS
During lab after finish lab exercise or Milner lab or DEG 17 (PRC)
Check the PRC hours:
Final Projects

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

4. 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

5. 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

6. 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

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

8. 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

9. 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

10. 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

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)?
Step 3 X Y A B C D E
15.20
SSX
16.0
SSY
14.0 SP
Example
Step 2
Step 1

12. 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’d expect by chance? D E
Example

13. 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’d expect by chance? D E
Example

14. 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

15. 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)
Generally, the variables correlated (they are not independent)
Hypothesis testing with Pearson’s r

16. Hypothesis testing with Pearson’s r
Step 1: Hypotheses
One -tailed
Hypothesize that variables are negatively correlated
Note: symbol ρ (rho)
is actually correct,
but rarely used
H0:
ρ ≥ 0
HA:
ρ < 0
Hypothesis testing with Pearson’s r

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

18. 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 X Y
2-tailed
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

19. 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

20. 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

21. 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

22. Example: New X Y 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

23. 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

24. Example: New X Y 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)? X Y
2-tailed
ρ = 0
H0:
HA:
ρ ≠ 0
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

25. 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

26. 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

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

28. SPSS: HGT.SAV Height by Weight, N = 40
Note that significance is expressed the same as previously
r (38) = .794, p < .01
What is p for 1-tailed test?
For df = 38, α = .05, 2-tailed, rcrit = .31
SPSS: HGT.SAV Height by Weight, N = 40

29. Minimum N = 30,
df = 28, rcrit = .30