T STATISTIC

t statistic

a statistic used for hypothesis testing when the population standard deviation is unknown.

z-score

a statistic used when the population standard deviation is known.

estimated standard error (SM)

an estimate of the real standard error (σM) when σ is unknown; computed from the sample variance or standard deviation; represents the standard distance between a sample mean (M) and the population mean (μ).

degrees of freedom (df)

the number of values in a sample that are free to vary; calculated as df = n

t distribution

a distribution used instead of the normal distribution when σ is unknown; becomes more like the normal distribution as df increases.

independent samples t-test

compares the means of two independent groups to determine if there is a significant difference between them.

repeated-measures t-test

compares means from the same group at two different times (before and after treatment) to evaluate the effect of an intervention.

matched-subjects design

each participant in one group is matched with a participant in another group on a specific variable to simulate repeated measures.

steps in computing t-test

(1) calculate sample variance or standard deviation, (2) calculate estimated standard error, (3) compute the t statistic.

pooled variance

a weighted average of two sample variances; used in independent samples t-test to compute estimated standard error.

hypothesis (null, H0)

a statement that there is no effect or no difference; tested in statistical analysis.

alternative hypothesis (H1)

a statement that there is an effect or difference.

two-tailed test

tests for any significant difference (either direction) between groups.

one-tailed test

tests for a difference in a specific direction.

critical region

the area in the tail(s) of the distribution that corresponds to a significance level (e.g., α = .05); if t falls here, H0 is rejected.

significance level (α)

the probability of rejecting the null hypothesis when it is actually true; commonly set at 0.05.

effect size

a measure of the magnitude of a treatment effect; provides information beyond significance.

Cohen's d

a measure of effect size indicating the standardized difference between two means.

r-squared (r2)

proportion of variance in the dependent variable explained by the independent variable or treatment; also used as an effect size.

percentage of variance explained

refers to the percent of total variation in scores that is accounted for by the treatment.

Hartley's F-max test

a test used to assess the homogeneity of variances assumption in an independent samples t-test.

homogeneity of variance

the assumption that two populations have equal variances; required for pooled variance computation.

assume equal variance (in software)

an option that tells software to assume the two groups have the same variance.

not assuming equal variance

a more conservative approach when performing a t-test in software.

order effects

changes in participants' performance due to the order in which treatments are administered (relevant in repeated-measures design).

counterbalancing

a method used to control for order effects by varying the order of treatments among participants.

time-related factors

external factors that may influence participants between treatment conditions in repeated-measures designs.

repeated-measures design

a design where the same participants are measured under different conditions or times.

independent-measures design (between-subjects)

a design with separate groups for each condition.

directional hypothesis

a hypothesis that specifies the direction of the expected difference.

non-directional hypothesis

a hypothesis that only states there will be a difference, without specifying the direction.

Excel t-test function

a built-in Excel tool to calculate t-test statistics, with options for equal or unequal variances.

reporting t-test results

include sample means (M), standard deviations (SD), t statistic value, degrees of freedom (df), p-value, and effect size.

example of reported results (independent t-test)

"Students in the dimly lit room (M = 12, SD = 3.07) scored significantly higher than those in the well-lit room (M = 8, SD = 2.93), t(14) = 2.67, p < .05, d = 1.33."

example of reported results (repeated-measures t-test)

"Experiencing a night of below-average sleep increased academic problems the next day by an average of M = 4.00 points (SD = 4.00), t(7) = 2.84, p < .05, r2 = 0.536."