μ(x̄1 - x̄2) = μ1 - μ2

$$\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$$

Estimate using sample standard deviations: $$\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

$$Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$$.

The degrees of freedom can be approximated using Welch's t-test formula, which is complex and often calculated by statistical software. A conservative approach is to use the smaller of n1-1 and n2-1.

Population SD (σ): True variability in the entire population. | Sample SD (s): Estimate of population variability based on the sample.

t-distribution: Used when population standard deviations are unknown and estimated by sample standard deviations, especially with small sample sizes. | z-distribution: Used when population standard deviations are known or when sample sizes are very large.

Single Mean: Distribution of sample means from one population. | Difference of Two Means: Distribution of the differences between sample means from two populations.

Independent Samples: Data from one sample does not influence the other. | Dependent Samples: Data from one sample is related to the other (e.g., matched pairs).

Parameter: A numerical value that describes a characteristic of the entire population (e.g., the true difference in population means). | Statistic: A numerical value that describes a characteristic of a sample (e.g., the difference in sample means).

The distribution of all possible differences in sample means (x̄1 - x̄2) that we would get if we repeated the study many times.

If sample sizes are large enough (typically n ≥ 30 for both samples), then the sampling distribution of the difference in sample means will be approximately normal, regardless of the shape of the original population distributions.

The average value of a variable in the entire population.

A measure of the spread or variability of data points around the population mean.

The average value of a variable calculated from a sample of the population.