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Reasoning in Psychology Using Statistics

Explore the strength and significance of relationships between variables using Pearson’s correlation in psychology. Learn to interpret the data, run SPSS, and conduct hypothesis testing effectively.

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Reasoning in Psychology Using Statistics

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  1. Reasoning in PsychologyUsing Statistics Psychology 138 2017

  2. Start working on your Final Projects soon (see link on syllabus page) • DueWed, May 3 (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: http://psychology.illinoisstate.edu/prc/hours.shtml Final Projects

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

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

  5. study time • test performance 115 mins 15 mins • 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: Relationships between variables

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

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

  8. Y 6 5 4 3 2 1 X 1 2 3 4 5 6 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)? A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 Example

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

  10. 2.4 2.0 4.8 4.0 5.76 -2.6 6.76 -2.0 4.0 5.2 1.4 1.96 2.0 4.0 2.8 -0.6 0.36 0.0 0.0 0.0 -0.6 0.36 -2.0 4.0 1.2 SP 3.6 4.0 0.0 15.20 0.0 16.0 14.0 SSX mean SSY Step 2 Step 1 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)? A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 Example

  11. Step 3 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)? A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 SP 15.20 16.0 14.0 SSX SSY Example Step 2 Step 1

  12. Y 6 5 4 3 2 1 X 1 2 3 5 4 6 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)? • Appears linear • Positive relationship • Fairly strong relationship • .898 is far from 0, near +1 A 6 6 B 1 2 C 5 6 • Fairly strong, but stronger than you wouldexpect by chance? D 3 4 E 3 2 Example

  13. 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 6 5 4 3 2 1 X 1 2 3 5 4 6 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)? A 6 6 B 1 2 C 5 6 • Fairly strong, but stronger than you would expect by chance? D 3 4 E 3 2 Example

  14. 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 A 6 6 B 1 2 C 5 6 D 3 4 E 3 2 Example

  15. 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

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

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

  18. H0: ρ =0 HA: ρ ≠ 0 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 There is no correlation between the study time and exam performance A 6 6 B 1 2 C 5 6 There is a correlation between the study time and exam performance D 3 4 E 3 2 Example: New

  19. 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. H0: ρ =0 HA: ρ ≠ 0 You decide to test whether there is any statistically significant relationship between these variables (alpha = 0.05). 2-tailed • Step 2 A 6 6 α = 0.05 B 1 2 C 5 6 D 3 4 E 3 2 Example: New

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

  23. Critical values of r (rcrit) • 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 Hypothesis testing with Pearson’s r

  24. Y 6 5 4 3 2 1 X 1 2 3 5 4 6 rcrit = ±0.878 H0: ρ =0 HA: ρ ≠ 0 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 df = n - 2 = 3 α = 0.05 • Step 5 A 6 6 From table B 1 2 C 5 6 D 3 4 E 3 2 Example: New

  25. rcritical Fail to Reject H0 Reject H0 -1.0 0.0 +1.0 • 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 Hypothesis testing with Pearson’s r

  26. Y 6 2-tailed 5 4 3 2 1 X 1 2 3 5 4 6 H0: r = HA: r ≠  -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)? df = n - 2 = 3 alpha = 0.05 A 6 6 rcrit = ±0.878 B 1 2 • Step 5 C 5 6 • Reject H0 • Conclude that the correlation is not equal to 0 D 3 4 E 3 2 “There is a significant positive correlation between study time and exam performance” Example: New

  27. 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

  28. SPSS: HGT.SAVHeight 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

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

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