t test

Students are required to demonstrate knowledge and understanding of inferential testing.
Familiarity with the use of inferential tests is essential.

Inferential Tests: These are statistical tests used when data is measured at the interval level.
- Interval Level Data: Data measured on a scale with equal intervals, which allows for the calculation of means and variances.
Power Comparison: Parametric tests (such as t-tests) are generally more powerful than non-parametric tests (such as Mann-Whitney and Wilcoxon tests).

Unrelated t-test: A statistical test used to determine if there is a significant difference between the means of two independent (unrelated) groups.

The unrelated t-test is employed when an independent groups design is used.
It tests the difference between two sets of data measured under interval conditions.

Scenario: Comparing the time taken to complete a jigsaw puzzle between boys and girls using an independent groups design.
Conditions: Boys and girls constitute the two different groups (the treatment is jigsaw completion).
Assumptions for Parametric Tests:
- Data is interval level (equal units).
- Normal distribution of the data from the population.
- Homogeneity of variance: standard deviations in both groups are similar.

Related t-test: A statistical test used for comparing means from the same group at different times (same participants).

The related t-test is applied when a repeated measures design is used.
It examines if there is a significant difference in the means of two related groups.

Scenario: Measuring heart rate before and after a treatment in the same participants.
Heart rate (measured in beats per minute) is also an example of interval level data based on a safe scale.
Assumptions: Normal distribution and homogeneity of variance for related designs.

Alternative Hypothesis: There is a difference in the time taken by boys and girls to complete a jigsaw puzzle.
Null Hypothesis: There is no difference in the time taken by boys and girls to complete a jigsaw puzzle.

Alternative Hypothesis: There is a reduction in heart rate activity when comparing heart rates before and after CBT treatment (directional, one-tailed).
Null Hypothesis: There is no difference in heart rate activity comparing heart rates before and after CBT.

Jigsaw Puzzle Study:
- Table 1 Data: Calculate sums and squares of scores for Group A (boys) and Group B (girls).
- Example Calculations:
  - Sum scores for Group A: $XA$
  - Sum scores for Group B: $XB$
  - Square each value in $XA$ and $XB$ and sum these squares.
Heart Rate Study:
- Table 3 Data: Calculate differences between scores for Condition A and Condition B.
- Steps:
  - Calculate the difference (d) between scores for Condition A (heart rate before treatment) and Condition B (heart rate after treatment).
  - Square each difference (d) and sum these squared differences.

Calculated Value of t: Measure used to determine significance in both tests.
- Jigsaw Puzzle: Calculated t value was $t = -0.116$ which indicates a lack of significance (p > 0.05), thus accepting the null hypothesis.
- Heart Rate: Calculated t value was $t = 2.237$; if this exceeds critical values from the t-distribution table, the null hypothesis can be rejected, indicating a significant difference in heart rates due to CBT treatment.

For Pairwise Comparison:
- Pearson's r: $df = N - 2$
- Related t-test: $df = N - 1$
- Unrelated t-test: $df = NA + NB - 2$
- Where $NA$ and $NB$ are the participant counts in two conditions.

For a larger sample, if a research design has larger groups (e.g., 61 boys and 61 girls), the strength of statistical tests is enhanced, improving the significance of results even with identical t-values calculated previously.

Application of concepts discussed, with examples and direction for hypothesizing.
Recognizing when to use dependent versus independent tests based on study design factors (e.g., repeated measures, independent groups).