Related t-Test and Unrelated t-Test
Related t-Test
- Purpose of Related t-Test
- Used when a repeated measures design is employed.
- Examines the difference in average heart rate before and after treatment (Cognitive Behavioral Therapy - CBT).
- Type of design: Repeated measures (same participants tested twice).
- Level of measurement: Interval, as heart rate (bpm) uses a scale that is publicly recognized and has equal intervals.
- Assumptions:
- Participants drawn from a normally distributed sample.
- Homogeneity of variance is assumed since it's a related design.
Study Context
- Study on Addiction
- Investigated the effects of CBT on physiological arousal among persistent gamblers.
- Sample size: 10 participants.
- Procedure:
- Participants played on a fruit machine for 20 minutes while heart rate was monitored (before treatment).
- Following a 6-week CBT course, the same participants played the same game for the same duration with heart rate monitored again.
Hypotheses
Alternative Hypothesis (H1):
- A reduction in heart rate activity is expected when comparing heart rates before and after CBT (directional, one-tailed).
Null Hypothesis (H0):
- No difference in heart rate activity when comparing heart rate before and after CBT.
Step 1: Data Calculations
Table 3: Calculations Table
| Participant | Condition A Heart Rate (bpm) | Condition B Heart Rate (bpm) | Difference (d) | d² |
|---|---|---|---|---|
| 1 | 84 | 80 | 4 | 16 |
| 2 | 71 | 70 | 1 | 1 |
| 3 | 52 | 55 | -3 | 9 |
| 4 | 66 | 58 | 8 | 64 |
| 5 | 58 | 58 | 0 | 0 |
| 6 | 77 | 70 | 7 | 49 |
| 7 | 63 | 61 | 2 | 4 |
| 8 | 81 | 75 | 6 | 36 |
| 9 | 71 | 74 | -3 | 9 |
| 10 | 70 | 61 | 9 | 81 |
| Σd | 31 | 269 |
Calculating the Value of t
Formula:
t = \frac{NΣd² - (Σd)²}{N - 1}
where:- N = number of pairs (here, N = 10)
- Σd = Sum of differences
- Σd² = Sum of squared differences
Plugging in the values:
t = \frac{10(269) - (31)²}{10 - 1}
t = \frac{2690 - 961}{9}
t = \frac{1729}{9}
t \approx 192.11- Final calculated t = 2.237
Step 2: Critical Values
- Calculated vs. Critical Values
- Calculated t value = 2.237
- Critical t value for one-tailed test at 0.05 significance level: 1.833 (degree of freedom: df = N-1 = 9)
- Since calculated t (2.237) > critical t (1.833), the result is significant.
- Conclusion: Reject the null hypothesis.
- Finding: There is a reduction in heart rate activity post-CBT (p < 0.05).
Methods: t-Tests and Taxi Drivers
- Reference Maguire et al. study for additional context on methods.
Unrelated t-Test
- Purpose of Unrelated t-Test
- Test of difference between two independent groups, using interval data.
- Assumptions required:
- Participants drawn from a normally distributed population.
- Homogeneity of variance exists (similar standard deviations).
Study Context
- Investigation into visuo-spatial ability differences in a jigsaw puzzle test between genders.
- Sample: 10 boys and 10 girls.
- Procedure: Completion of a jigsaw puzzle, time taken recorded for comparison.
Hypotheses
Alternative Hypothesis (H1):
- A difference exists in time taken by males and females to complete the jigsaw puzzle (non-directional, two-tailed).
Null Hypothesis (H0):
- No difference in time taken by males and females.
Step 1: Data Calculations
Table 1: Calculations Table
| Participant | Group A Time Taken (sec) | XA² | Group B Time Taken (sec) | XB² |
|---|---|---|---|---|
| 1 | 64 | 4096 | 1 | 52 |
| 2 | 56 | 3136 | 2 | 59 |
| 3 | 89 | 7921 | 3 | 90 |
| 4 | 55 | 3025 | 4 | 112 |
| 5 | 79 | 6241 | 5 | 84 |
| 6 | 102 | 10404 | 6 | 73 |
| 7 | 80 | 6400 | 7 | 79 |
| 8 | 69 | 4761 | 8 | 64 |
| 9 | 69 | 4761 | 9 | 49 |
| 10 | 80 | 6400 | 10 | 90 |
| ΣXA = 743 | ΣXA² = 57145 | ΣXB = 752 | ΣXB² = 60052 |
Calculating the Value of t
Formula:
t = \frac{(XA - XB)}{\sqrt{\frac{SA^2}{NA} + \frac{SB^2}{NB}}}
Where:- $XA$, $XB$ are the means of groups A and B.
- $SA$ and $SB$ are the standard deviations for groups A and B respectively.
- $NA$ and $NB$ represent number of participants in groups A and B respectively.
Mean Calculation:
- Sum of scores:
- From Group A: $EX_A$ (743)
- From Group B: $EX_B$ (752)
- Example:
a. SA = \sqrt{\frac{ΣXA^2 - \frac{(ΣXA)^2}{NA}}{NA-1}} b. SB = \sqrt{\frac{ΣXB^2 - \frac{(ΣXB)^2}{NB}}{NB-1}}
Step 2: Critical Values
- Calculated vs. Critical Values
- Critical t value (two-tailed test at 0.05 significance level; df = NA + NB - 2 = 18) = 2.101.
- Calculated t value (as per final calculations) = -0.116
- Since absolute value of calculated t < critical t, the result is not significant.
- Conclusion: Accept the null hypothesis: No difference between males and females in jigsaw puzzle completion time (p > 0.05).
Summary of Important Points
Parametric Tests Typically Include:
- Related t-test (repeated measures)
- Unrelated t-test (independent measures)
Inferential Tests Understanding
- Necessary to understand the differences and applications of both tests.
- Acknowledge less powerful non-parametric tests available, such as the Mann-Whitney and Wilcoxon tests, but prefer t-tests for interval data when assumptions met.