Lec9: The t-Distribution and t-Tests

Vocabulary practice flashcards covering fundamental and advanced statistical concepts including the t-distribution, t-test variants, assumptions, degrees of freedom, Levene's test, and Cohen's d effect sizes.

One Sample t-test

A statistical test used to compare a sample mean to a known population mean when the population standard deviation ($SD$) is unknown.

One Sample z-test

A statistical hypothesis test used to compare a sample mean to a known population mean when the population standard deviation ($SD$) is known.

Independent Samples t-test

A parametric test used to determine whether the means and underlying distributions of two unrelated, independent groups are statistically significantly different from one another.

Paired t-test

A statistical test used when two groups are related, such as in repeated measures or pretest-posttest designs where the same subjects are measured twice.

Homogeneity of Variance

An assumption of the independent samples $t$-test requiring that the variances of the two comparison groups are not significantly different from each other.

Robustness

The capacity of a statistical test to produce valid results even when one or more of its underlying assumptions are violated.

William Sealy Gosset

The chemist, statistician, and chief brewer at Guinness ($1876$–$1937$) who developed Student's $t$-distribution to detect differences in crop yields with small sample sizes.

Student

The pen name used by William Sealy Gosset when publishing his $1908$ paper, "The Probable Error of a Mean," in the journal Biometrika.

Degrees of Freedom ($df$)

The number of values in a final statistical calculation that are free to vary; for an independent samples $t$-test, it is calculated as $n_1 + n_2 - 2$.

Pooled Variance

The weighted average variance of two groups combined, which forms the denominator of the independent samples $t$-test: $\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$.

Critical Value of t ($t_{cv}$)

The threshold value obtained from a $t$-distribution table based on degrees of freedom ($df$) and significance level ($\alpha$); if obtained $t > t_{cv}$, the test is statistically significant ($p < .05$).

Levene's Test for Equality of Variances

An inferential test assessing whether group variances are equal; a non-significant result allows reading the "Equal variances assumed" row, while a significant result requires reading the "Equal variances not assumed" row.

Cohen's d

A standardized effect size measure for independent $t$-tests calculated by dividing the difference between group means by the average standard deviation: $\frac{M_1 - M_2}{(SD_1 + SD_2)/2}$.

Small Effect Size (Cohen's Guideline)

A benchmark standard for effect size indicated by a Cohen's $d$ value of approximately $0.2$.

Medium Effect Size (Cohen's Guideline)

A benchmark standard for effect size indicated by a Cohen's $d$ value of approximately $0.5$, considered practically useful and worth the trouble.

Large Effect Size (Cohen's Guideline)

A benchmark standard for effect size indicated by a Cohen's $d$ value of approximately $0.8$.

Statistical Power

The probability of correctly detecting a statistically significant group difference; it increases with larger sample sizes, larger effect sizes, smaller standard deviations, and directional (one-tailed) tests.

Neil Salkind's 3 Principles

Three guidelines for contextualizing statistical significance: ensure theory and measures are sound, confirm benefits outweigh costs using effect size, and report when treatments fail to work.

Normality Assumption (Independent t-Test)

The assumption that the population from which samples are drawn is normally distributed; the independent $t$-test is robust to its violation if $n \ge 30$.

PICO Question

A formalized research question structure used in clinical literature reviews to define the Patient/Population, Intervention, Comparison, and Outcome.