Lecture 8: Paired-Samples t Test

Vocabulary and key concepts from the Day 8 lecture on Paired-Samples t Tests, including formulas for standard deviation, standard error, and effect size.

z Test Parameters

The parameters that must be known for a z test, but not a single-sample t test, are BOTH the population mean ($\mu$) and the population standard deviation ($\sigma$).

Corrected Sample Standard Deviation ($s$)

The statistic used to estimate the population standard deviation in a single-sample t test, calculated with the formula $s = \text{sq rt}\frac{\sum_{i=1}^{N} (X - M)^2}{N - 1}$.

Degrees of Freedom ($df$)

Calculated as $N - 1$ for a single-sample t test, this value is necessary to determine which specific t distribution to use among multiple available distributions.

Paired-Samples t Test

A statistical test appropriate when observations in two sample groups are paired, such as when each participant is observed twice (e.g., before and after a treatment).

Difference Score

The value calculated by directly comparing the difference across the two observations for each individual participant.

Distribution of Mean Differences

The relevant comparison distribution for a paired-samples t test, used instead of a distribution of means because each observation is paired.

$\mu_M$ (Mean of the Comparison Distribution)

In a paired-samples t test, this value is $0$ based on the null hypothesis assumption that there is no difference between the sample means.

Standard Error ($s_M$) for Paired Samples

The approximation of the standard error of the mean difference, calculated using the formula $s_M = \frac{s}{\sqrt{N}}$.

t-statistic (Paired-Samples)

The test statistic calculated using the formula $t = \frac{M_{Difference} - \mu_M}{s_M}$.

95% Confidence Interval for Paired Samples

A range defined by a Lower bound of $M_{difference} - t_{critical}(s_M)$ and an Upper bound of $M_{difference} + t_{critical}(s_M)$, using difference scores.

Cohen's d (Paired-Samples)

A measure of effect size for paired samples calculated using the formula $\text{Cohen's d} = \frac{M - \mu}{s}$, where values concern the difference scores.

Within-Groups Design

Also known as a Within-Subjects Design, this is the research design for which a paired-samples t test is the appropriate statistical analysis.