03 CARDS Group DIfference t-Test

Group Differences in Psychological Statistics

Types of Samples

• Dependent Samples

  • Also known as matched pairs and repeated measures

  • Same subjects participate in both conditions of the experiment

• Independent Samples

  • A.k.a independent measures

  • Different groups of subjects perform different conditions of the experiment

Variables

• Both dependent and independent samples involve one independent variable (IV) with two levels (conditions).

• One dependent variable is measured as an outcome.

The Logic of t-Tests

t-Statistic Formula

• The t-statistic is expressed as:

  • [ t = \frac{(M_1 - M_2) - (\mu_1 - \mu_2)}{S_{(M_1 - M_2)}} ]

  • Where:

    • Numerator: Difference between sample means and hypothesized population means

    • Denominator: Estimated standard error of the difference between sample means

• Similar in logic to z-scores.

Understanding the t-Test

• Definition: A parametric test comparing differences for two groups or scores.

Assumptions of the t-Test

1. Continuous Dependent Variable

2. Categorical Independent Variable with 2 levels

3. Independence of observations

4. Normal distribution of the data

5. Homogenous variance between groups (homoscedasticity)

6. At least 6 subjects per group for a balanced design

Alternatives to t-Test

• Wilcoxon Signed-Rank Test (for unequal variance)

• Mann-Whitney U Test

Choosing the Right t-Test

Dependent-Samples t-Test

• Used for paired samples from the same population.

• Design: Repeated Measures Design

• Example: Same people tested before and after treatment.

Independent-Samples t-Test

• Used for two different groups (between-subjects design).

• Example: Completely different samples.

More on Independent-Samples t-Test

Characteristics

• Used when there is no prior knowledge about population parameters (μ, σ).

• Estimates variance using sample data.

Key Focus

• Evaluates if sample means come from the same population or two different populations.

t-Test Types

One-Sample t-Test

• Compares one group to a known standard (whether a group's mean differs from a standard value).

One-Tailed vs. Two-Tailed Tests

• One-Tailed Test: Hypothesis specifies direction (e.g., one group higher than the other).

• Two-Tailed Test: Tests for any significant difference, regardless of direction.

Hypothesis Formulation

Null Hypothesis

• Assumes no significant difference between population means.

Alternative Hypothesis

• Suggests a significant difference does exist.

Degrees of Freedom

• Calculated as follows:

  • One-Sample t-Test: df = n - 1

  • Dependent Samples: df = n - 1 (paired measurements)

  • Independent Samples: df = n1 + n2 - 2

Effect Size Measures

Cohen’s d

• Standardized measure of mean difference

• Formula: [ d = \frac{M_1 - M_2}{s} ]

• Compares observed differences to expected random differences.

R-Squared (R²)

• Measures variability in scores explained by treatment effects.

Reporting Results in APA Format

• Example Structure:

  • Results indicate a significant difference between noise group (M=3.45, SD=1.11, n=12) and no-noise group (M=3.00, SD=0.80, n=12), t(22) = 4.00, p = .001. Effect size d = .002.

• Important to note: If p < .001, exact value may not need to be stated.

Group Differences in Psychological Statistics

Types of Samples

Dependent Samples

• Example: A study measuring the effect of a new study technique on test scores, where the same group of students takes a pre-test before using the technique and a post-test afterward to evaluate improvement.

Independent Samples

• Example: A clinical trial comparing the effects of two different medications on blood pressure, with one group receiving medication A and another group receiving medication B

Variables

• Example: In a study examining the impact of sleep on cognitive performance, the independent variable is the amount of sleep (group with 6 hours vs. group with 8 hours), and the dependent variable is the performance score on a cognitive test.

The Logic of t-Tests

t-Statistic Formula

• Example: In a study measuring the difference in average height between two samples of plants, the t-statistic helps determine if the difference in mean heights is statistically significant.

Understanding the t-Test

Definition

• Example: When comparing test scores of students from two different teaching methods, a t-test can help identify whether the mean difference in scores is significant.

Assumptions of the t-Test

• Example: A researcher tests whether a new reading program affects the scores of 2nd graders and assumes that the scores are normally distributed, the students are independent, and there are equal variances between the two teaching methods.

Choosing the Right t-Test

Dependent-Samples t-Test

• Example: A researcher measures blood glucose levels in diabetic patients before and after a new diet plan.

Independent-Samples t-Test

• Example: A study evaluating the effectiveness of two different exercise programs on weight loss; participants in one group follow Program A, while another group follows Program B.

More on Independent-Samples t-Test

Characteristics

• Example: Researchers evaluating the test scores of two different classes using different textbooks for English, without prior assumptions of the classes’ performance.

Key Focus

• Example: Investigating whether students in a private school outperform those in a public school on a standard math assessment.

t-Test Types

One-Sample t-Test

• Example: A school compares its average student GPA to the national average GPA to see if they are significantly different.

One-Tailed vs. Two-Tailed Tests

• Example: A researcher tests whether a new drug significantly lowers cholesterol levels (one-tailed) versus simply testing if it has any effect on cholesterol (two-tailed).

Hypothesis Formulation

Null Hypothesis

• Example: "There is no significant difference in average test scores between Group A and Group B."

Alternative Hypothesis

• Example: "There is a significant difference in average test scores between Group A and Group B."

Degrees of Freedom

Calculated as follows:

• Example: A study with one sample of size 30 would have degrees of freedom calculated as df = n - 1 = 29.

Effect Size Measures

Cohen’s d

• Example: In a clinical trial where the mean difference in recovery times between two therapies is measured, Cohen’s d can quantify the difference’s size.

R-Squared (R²)

• Example: In a regression analysis on the effect of study hours and sleep on GPA, R² indicates how much variability in GPA is explained by the study hours and sleep combined.

Reporting Results in APA Format

• Example: "Results indicated a significant difference in test scores between the experimental group (M=88, SD=5.5, n=15) and control group (M=75, SD=6.0, n=15), t(28) = 3.92, p < .001, Effect size d = 0.85."