Bivariate Data Analysis 1: Paired and Independent Samples T-Tests

Course Roadmap

• Phase 1: Measurements, Experiment Designs, Hypothesis Testing, Exam 1

• Phase 2: t-test, ANOVA, Exam 2

• Phase 3: Correlation & Regression, Chi-Square & Logistic Regression, Validity, Exam 3

Fundamentals of Bivariate Analysis

• Definition: Evaluates relationships between 2 variables or compares 2 distinct groups.

Within-Group vs. Between-Group

• Within-Group (Paired Samples):

  • 1 group, 2 variables/data points per person.

  • Test: Paired samples t-test.

• Between-Group (Independent Samples):

  • 2 groups, 1 variable/data point per person.

  • Test: Independent samples t-test.

Signal-to-Noise Framework

• Core Formula:   $\text{Test Statistic} = \frac{\text{Signal}}{\text{Noise}} = \frac{\text{Mean Difference}}{\text{Standard Error}}$   $t = \frac{\bar{x}_1 - \bar{x}_2}{\text{SE}}$

• Principles:

  • High Signal + Low Noise $\rightarrow$ Large $t$ (Statistically Significant).

  • High Noise (high variability/SE) $\rightarrow$ Small $t$ (Not Significant).

Paired Samples T-Test (JASP)

• Original ($B1\_I$) vs. Digital Display 1 ($C1\_I$):

  • $N = 300$, $t(299) = -20.027$, $p < .001$ (Significant).

• Original ($B1\_I$) vs. Digital Display 2 ($D1\_I$):

  • $t(299) = -1.247$, $p = .107$ (Not significant).

• Tails:

  • One-Tailed: Directional hypothesis ($\alpha = .05$ in 1 tail).

  • Two-Tailed: Non-directional ($\alpha = .05$ split into $.025$ per tail).

APA 7th Style Rules

• Italicize symbols: $p$, $M$, $SD$, $t$, $df$, $N$, $n$.

• Format: $t(df) = \text{value}, p = \text{value}$ (spaces around relational signs).

• Decimals: Omit leading zero if statistic cannot exceed 1; include zero if it can exceed 1.

• No commas in degrees of freedom.

Independent Samples T-Test Formulas

• Equal Group Sizes ($n_1 = n_2$):   $\text{SE}_{\text{diff}} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

• Unequal Group Sizes ($n_1 \neq n_2$):   $s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}$   $\text{SE}_{\text{diff}} = \sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}}$

• t-Score:   $t = \frac{\bar{x}_1 - \bar{x}_2}{\text{SE}_{\text{diff}}}$

Independent Samples T-Test (JASP)

• Prior Usage ($A1\_N$) & NPS ($B1\_I$):

  • $t(298) = 12.28$, $p < .001$ (Significant).

• Pink Preference ($A2\_N$) & NPS ($B1\_I$):

  • $t(298) = -1.368$, $p = .172$ (Not significant).

Practical Application

• Test grouping variable ($A5\_N$) vs. rating ($B2\_I$).

• Run independent samples t-test and report $t(df)$, $p$, group means, and SEs in APA format.