Paired Samples T-Test Notes

Paired Samples T-Test

Course Objectives

Distinguish between different types of t-tests:
- One-Sample T-Test
- Independent-Sample T-Test
- Paired-Sample T-Test
Describe the usage and rationale for a paired-sample t-test.
Compute and interpret the paired-sample t statistic.
Explain how difference scores are used in the calculation of t.
Interpret the relationships among t, p, and effect size.

One-Sample T Revisited

Purpose: To compare the mean of one group against a specific known population value.
Formula:
$t = \frac{X - \mu_0}{s / \sqrt{n}}$
Standard Error (SE):
$SE = \frac{s}{\sqrt{n}}$
Simplified Formula:
$t = \frac{X - \mu_0}{SE}$
Context: Used in scenarios where the population standard deviation (SD) is unknown.
Example: Do psychology students’ average stress scores differ from the university's average?

One Sample and Population vs. Multiple Samples

One-Sample T-Test: Compares a sample mean against a single population value.
Multiple Samples Tests: Involves comparing two or more means (e.g., before vs. after measures, or comparisons between multiple groups).

Paired Samples vs. One-Sample T

One-Sample Example: Compares average sleep hours of UAH students to a national average.
Paired-Sample Example: Compares the same students’ sleep hours before and after finals week.
Key Distinction: The paired t-test uses two related measurements from the same participants.

Between vs. Within-Subjects Design

Between-Subjects Design: Different participants are involved in each condition.
Within-Subjects Design: The same participants are measured multiple times across conditions.

Between-Subjects Design

Example: Comparing exam scores between students from two different classes.
Approach: Each participant provides only one score, requiring the use of an independent-samples t-test.

Within-Subjects Design

Example: Comparing exam scores for the same class before and after tutoring sessions.
Approach: Same participants are measured twice, necessitating the use of a paired-samples t-test.

Paired vs. Independent Samples T

Independent Samples T-Test: Compares means from different groups containing different individuals.
Paired Samples T-Test: Compares means from the same or matched individuals within the same group.

Check Our Understanding

Identify Tests:
1. Pre/post stress for after meditation → use paired-sample t-test.
2. Stress levels between two companies → use independent-sample t-test.
3. A sample’s stress level vs. national average → use one-sample t-test.

Paired Samples T-Test

Designed for dependent or related samples.
Known as within-subjects or repeated-measures t-test.
Example: Memory performance before and after caffeine consumption.

Basic T-Test Formula

General Logic: $t = \text{effect} / \text{error}$
- Mean difference divided by standard error.

Repeated Measures T Formula

Numerator: Mean difference between two time points (e.g., Time 1 vs. Time 2).
Denominator: Variability in difference scores, indicating how consistent individual changes are.

Full Mathematical Formula

$t = \frac{X1 - X2}{SE_{diff}}$
Where:
- $X1 - X2$ = mean difference between time points
- $SE_{diff}$ = standard error of the difference scores

Calculating the Denominator Steps

Compute the Sum of Squared Differences (SSdiff).
Calculate Standard Deviation of the differences (Sdiff).
Compute Standard Error of the Differences (SEdiff).

SSdiff Detailed Steps

Calculate each person’s difference.
Find the average difference across participants.
Subtract each person’s difference from the average difference.
Square each result.
Sum the squares.

Degrees of Freedom (df)

Calculated as:
$df = n - 1$
One degree of freedom is lost when estimating the mean difference.

Calculation of Sdiff

Take SSdiff.
Divide by df:
$S{diff} = \frac{SS{diff}}{df}$
Take the square root of the result.

Calculation of SEdiff

Calculated as:
$SE{diff} = \frac{S{diff}}{\sqrt{n}}$

Simplified Paired T Formula

Formula:
$t = \frac{X1 - X2}{SE_{diff}}$
Emphasis on using this formula for exams—no need to compute SSdiff or df manually.

Paired T Example 1

Mean Before: 70
Mean After: 75
SEdiff: 2.5
Calculation:
$t = \frac{75 - 70}{2.5} = 2.00$

Paired T Example 2 (Student Practice)

Mean Before: 60
Mean After: 67
SEdiff: 3.5
Task: Calculate t-value.

Paired T and P-Values

Use the calculated t-value to find the corresponding p-value.
Decision Logic:
- If p < 0.05 → reject the null hypothesis (H₀).
- If $p ≥ 0.05$ → fail to reject the null hypothesis (H₀).

Interpretation Example

Scenario: Knowledge tested before and after training.
Results: $t(19) = 2.4, p = 0.025$
Conclusion: Significant improvement in knowledge post-training; the null hypothesis is rejected.

Interpretation Example 2

Scenario: Performance difference before and after training.
Results: $t(15) = 1.2, p = 0.24$
Conclusion: No significant difference; training did not affect performance.

Interpretation Example 3 (Student Practice)

Scenario: Satisfaction difference before and after training.
Results: $t(18) = 2.6, p = 0.018$
Task: Determine significance and implications of results.

SPSS Output Interpretation

Paired Samples Test

Table Contents:
- Paired Differences:
- Significance Level
- 95% Confidence Interval of the Difference
- Example for Comparison:
- Pair 1:
 - Mean Difference: -4.63651
 - Std. Deviation: 2.79331
 - Std. Error Mean: .50999
 - Confidence Interval: Lower -5.67955, Upper -3.59347
 - t: -9.091
 - df: 29
 - One-Sided p: < .001
 - Two-Sided p: < .001
- Another Example:
- Pair 1 (Meditation):
 - Mean: -0.43925
 - Std. Deviation: 2.72647
 - Std. Error Mean: .49778
 - Confidence Interval: Lower -1.45732, Upper .57883
 - df: 29
 - One-Sided p: .192
 - Two-Sided p: .385

Final Note

Summary: This document covers the paired-samples t-test comprehensively, including its applications, calculations, example scenarios, and interpretation of results from statistical analysis.