Study Notes on Paired Samples T-Test

Paired Samples T-Test

Course Objectives

• Differentiate between t-tests:
  • One-sample t-test
  • Independent-sample t-test
  • Paired-sample t-test
• Purpose of Paired-Sample T-Test:
  • Used when comparing two related samples or measurements.
• Computation:
  • Calculate and interpret the paired-sample t statistic.
• Difference Scores:
  • Explanation of how difference scores are utilized to derive the t statistic.
• Relationship Between t, p, and Effect Size:
  • Interpret how these statistics interconnect and affect the analysis results.

One-Sample T-Test Overview

• Purpose:
  • To compare the mean of one group against a known population value.
• Formula:$t = \frac{X - \mu_0}{s / \sqrt{n}}$
  • Where:
  • $X$: Sample mean
  • $\mu_0$: Known population mean
  • $s$: Sample standard deviation
  • $n$: Sample size
• Standard Error (SE):
  $SE = \frac{s}{\sqrt{n}}$
• Simplified Formula:
  $t = \frac{X - \mu_0}{SE}$
• Application:
  • Example Question: Do psychology students’ average stress scores differ from that of the university average?

Types of Samples and T-Tests

• One-Sample:
  • Compares a sample mean to a specific population mean.
• Multiple Samples:
  • Compares two or more means, i.e., before vs. after scenarios, or comparisons between Group A and Group B.

Distinction Between One-Sample and Paired Samples T-Tests

• One-Sample Example:
  • Compare average sleep hours of UAH students with the national average.
• Paired-Sample Example:
  • Compare the same students’ sleep hours before and after finals week (addresses changes in the same subjects).

Between vs. Within-Subjects Design

• Between-Subjects Design:
  • Different individuals participate in each experimental condition.
  • Example: Comparing exam scores among different classes.
  • Requires independent-samples t-test.
• Within-Subjects Design:
  • The same individuals are measured multiple times (e.g., before and after an intervention).
  • Example: Exam scores of the same students before and after tutoring.
  • Requires paired-samples t-test.

Comparison of T-Tests

• Independent T-Test:
  • Utilizes different participants to compare means across groups.
• Paired T-Test:
  • Involves the same or matched participants to compare their measurements within subjects.

Understanding the Paired Samples T-Test

• Definition:
  • Utilized for dependent/related samples, also known as within-subjects or repeated-measures t-test.
• Example Scenario:
  • Evaluation of memory performance before vs. after caffeine consumption.

Basic T-Test Formula

• General Logic of T-Test:
  • $t = \frac{\text{effect}}{\text{error}}$
  • The mean difference divided by the standard error.

Repeated Measures T Formula

• Numerator:
  • Represents the mean difference between two measurements (e.g., Time 1 vs. Time 2).
• Denominator:
  • Measures the variability in the difference scores (indicating how consistent those changes are).

Full Mathematical Formula for Paired T-Test

• $t = \frac{X<em>1 - X</em>2}{SE_{diff}}$
• Components:
  • $X1 - X2$: Mean difference between two time points.
  • $SE_{diff}$: Standard error of the difference scores.

Steps to Calculate the Denominator

1. Calculate Sum of Squared Differences (SSdiff):
   • Collect the differences of each participant’s measurements.
2. Calculate Standard Deviation of Differences (Sdiff):
3. Calculate Standard Error of Differences (SEdiff):
   • $SE<em>{diff} = \frac{S</em>{diff}}{\sqrt{n}}$

Sum of Squared Differences (SSdiff) Calculation Steps

1. Calculate the difference for each participant.
2. Determine the average difference across all participants.
3. Subtract the average difference from each participant’s difference.
4. Square the results of these differences.
5. Sum the squared results to obtain SSdiff.

Degrees of Freedom (df)

• Calculation:$df = n - 1$
  • One degree is lost for estimating the mean difference.

Standard Deviation of Differences (Sdiff) Calculation Steps

1. Take SSdiff.
2. Divide by df.
3. Take the square root of the result to find Sdiff.
   • $S<em>{diff} = \sqrt{\frac{SS</em>{diff}}{df}}$

Standard Error of Differences (SEdiff) Formula

• $SE<em>{diff} = \frac{S</em>{diff}}{\sqrt{n}}$

Simplified Paired T Formula

• The simplified calculation:
  $t = \frac{X<em>1 - X</em>2}{SE_{diff}}$
• Note: For examinations, primarily focus on this formula without manually computing SSdiff or df.

Paired T-Test Example 1

• Given Data:
  • Mean before treatment: 70
  • Mean after treatment: 75
  • SEdiff: 2.5
• Calculation:
  $t = \frac{75 - 70}{2.5} = 2.00$

Paired T-Test Example 2 (Practice)

• Given Data:
  • Mean before treatment: 60
  • Mean after treatment: 67
  • SEdiff: 3.5
• Task: Calculate t.

Relationship Between Paired T and P-Values

• Use the calculated t to determine the p-value.
• Decision Logic:
  • If $p < .05$: reject null hypothesis (H₀).
  • If $p \geq .05$: fail to reject null hypothesis (H₀).

Interpretation Examples

• Example 1:

  • Analysis of knowledge before and after training.
  • Results:
    $t(19) = 2.4, p = .025$
  • Conclusion: Statistically significant improvement in knowledge after training; null hypothesis rejected.

• Example 2:

  • Assessing performance before and after training.
  • Results:
    $t(15) = 1.2, p = .24$
  • Conclusion: No statistically significant difference in performance; training did not lead to significant changes.

• Example 3 (Practice):

  • Evaluating satisfaction before and after training.
  • Hypothetical Result:
    $t(18) = 2.6, p = .018$
  • Task: Assess significance and interpret the implications of the result.

SPSS Output Interpretation

• Scenario Considered:
  • Paired Samples Test on treated vs. untreated groups.
• Table Includes:
  • Paired Differences:
  • Mean: -4.63651
  • Standard Deviation: 2.79331
  • Standard Error Mean: 0.50999
  • Confidence Interval: Lower: -5.67955, Upper: -3.59347.
  • Calculated t: -9.091 with df: 29, p < .001.
• Additional Test Example:
  • Before Meditation vs. After Meditation:
  • Mean Difference: -0.43925
  • Standard Deviation: 2.72647
  • Standard Error Mean: 0.49778.
  • Confidence Interval: Lower: -1.45732, Upper: 0.57883.
  • Calculated p: One-Sided: 0.192, Two-Sided: 0.385.

Conclusion on Paired Samples T-Test

• The Paired Samples T-Test is a crucial statistical method for analyzing the differences in means from the same participants in two different conditions, primarily addressing the within-subjects designs. It allows for insights into the effects of interventions or temporal changes on various dependent measures.