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.

It represents the distribution of differences in sample means (x̄1 - x̄2) from repeated samples. It allows us to make inferences about the true difference in population means (μ1 - μ2).

The CLT allows us to assume the sampling distribution of the difference in sample means is approximately normal when sample sizes are large (n ≥ 30 for both samples), even if the populations are not normally distributed.

Variances are added because they represent the squared deviations from the mean, and adding them combines the variability from both distributions. Taking the square root then returns the combined standard deviation.

Larger sample sizes decrease the standard deviation of the sampling distribution, leading to more precise estimates of the true difference in population means.

Checking conditions ensures that the sampling distribution is approximately normal, which is a requirement for using normal-based statistical tests. Failing to check can lead to invalid conclusions.

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.