Overview of Cohen's d
Cohen's d is a measure used to indicate the standardised difference between two means.
Formula:
d = \frac{\text{Sample Mean} - \text{Population Mean}}{\text{Population Standard Deviation}}
Application: Used in the context of a z-test where the population standard deviation is known.
Effect Sizes According to Cohen's d
Small Effect Size:
Defined as a Cohen's d of 0.2 or less.
Small to Medium Effect Size:
Cohen's d ranges from 0.2 to 0.5.
Medium to Medium-Large Effect Size:
Cohen's d from 0.5 to 0.8.
Large Effect Size:
Cohen's d of 0.8 and above.
Very Large Effect Size:
Cohen's d of 1.2 is also noted.
Large effect sizes denote practical significance and effectiveness of interventions or treatments.
Interpretation of Effect Sizes
It's important to contextualize effect sizes in the literature to determine practical versus statistical significance.
Example: If an intervention yields a Cohen's d of 0.42, which is below the medium effect size cutoff, it may still be significant if compared to other interventions with lower effect sizes (e.g., 0.27).
Z Tests
What is a Z Test?
A statistical test for comparing the sample mean to the population mean to determine if there is a significant difference between the two.
Requirements for a Z Test:
Knowledge of population mean and standard deviation.
Sample size must be 30 or greater (based on the Central Limit Theorem).
Z Critical Value and Alpha Level
Z critical value for a significance level (alpha) of 0.05 is approximately ±1.96.
Relationships:
As the alpha level decreases, the critical value increases, making it more difficult to reject the null hypothesis.
Alpha Level 0.01: critical value becomes approximately ±2.58.
Alpha Level 0.001: critical value of about ±3.3.
Risks and Errors in Hypothesis Testing
Type I Error (False Positive): Rejecting a true null hypothesis.
Type II Error (False Negative): Failing to reject a false null hypothesis.
Lowering alpha increases the risk of Type II error.
Statistical Decision-Making Framework
Defining Extreme Values:
Extreme values correspond with critical values to determine significance.
Explanation of Alpha Level:
At alpha 0.05, 2.5% of scores lie at either tail of the distribution, indicating significance if the sample mean lies beyond these points.
Steps in the One Sample Z-Test
State the Hypotheses:
Null Hypothesis (H₀): No treatment effect. For example, mean production with music = 80 chairs/day.
Alternative Hypothesis (H₁): Treatment effect present. Mean production with music ≠ 80 chairs/day.
Set the Criteria for Beginning the Test:
Select alpha level (e.g., 0.05) and determine critical Z values (±1.96 for alpha 0.05).
Collect Data and Calculate Z Statistic:
Calculate the mean of the sample, the standard error (SE = \frac{\sigma}{\sqrt{n}}), and the Z statistic using Z = \frac{\bar{X} - \mu}{SE}.
Interpret Results:
Compare calculated Z statistic to critical Z values. Make decisions whether to retain or reject the null hypothesis.
Example: Chair Production Study
Population Parameters:
Mean: 80 chairs/day
Population Standard Deviation: 9.35 chairs
Sample Data:
Sample Size: 30 chairmakers
Treatment: Listening to music
Sample Mean: 81.4 chairs/day
Calculating Standard Error
SE = \frac{9.35}{\sqrt{30}} \approx 1.71
Z Statistic Calculation
Z = \frac{81.4 - 80}{1.71} \approx 0.819
Decision Making
Since 0.819 is within the range of ±1.96, we fail to reject the null hypothesis, indicating no significant difference in chair production with music.
Conclusion
No significant statistical difference was determined from the intervention of music on chair production, leading to the conclusion that it does not significantly affect productivity for this sample.