Week 7: Understanding T-Tests in Statistics

One-Sample T-Tests

Used to compare the mean of a single sample to a hypothesized population mean.

Standard Scores (Z-scores)

(raw score - population mean) / standard deviation. All Z-scores have mean 0 and SD 143.

Diagnostic Test Statistic (T)

Measures the difference between a sample mean and a hypothesized population mean; should be close to zero if the null is true.

Sampling Distribution

If the null is true, T follows a t-distribution.

Effect Size (Cohen's d)

Interpretation: 0.2 (small), 0.5 (medium), 0.8 (large).

Independent Samples T-Tests

Compares the means of two independent groups.

Assumptions for Independent Samples T-Test

Population distributions are normal, observations are independently sampled, groups have the same standard deviation.

Student's T-test

Assumes equal variances; estimates a single 'pooled' standard deviation.

Welch's T-test

Does not assume equal variances; uses both groups' standard deviations separately; generally better and is the default in R.

Paired Samples T-Tests

Compares two means within a repeated measures design (e.g., before-after measurements on the same individuals).

How Paired Samples T-Test Works

Essentially runs a one-sample t-test on the difference scores (Null hypothesis: mean difference is 0).

T-Test Assumptions and Solutions

Data must be independent, except for paired designs.

Homogeneity of Variance

The two groups are drawn from populations with the same variance.

QQ Plots

Scatterplot of observed quantiles against theoretical normal quantiles; a straight line suggests normality.

Shapiro-Wilk Test

Quantifies departures from normality. A p-value < 0.05 suggests non-normality.

Non-Parametric Tests

Avoid assumptions about distribution shape; less powerful but sometimes necessary.

Independent Samples Wilcoxon Test

Non-parametric alternative to independent t-test.

Paired Samples Wilcoxon Test

Non-parametric alternative to paired t-test.

Effect Size for Wilcoxon

Cohen's d assumes normality, so use wilcox_effsize() from rstatix package.