$\bar{x}_1 - \bar{x}_2$

$t^* \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Point Estimate ± Margin of Error

Use a conservative estimate by taking the smaller of $n_1 - 1$ and $n_2 - 1$.

$(\bar{x}_1 - \bar{x}_2) \pm t^* \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

The best guess for the true difference in population means, calculated as $\bar{x}_1 - \bar{x}_2$.

The amount added and subtracted from the point estimate to create the confidence interval.

If sample sizes are at least 30 (n ≥ 30), the sampling distributions will be approximately normal.

When sampling without replacement, the sample size should be less than 10% of the population size.

The value from the t-distribution used in calculating the margin of error, based on the confidence level and degrees of freedom.

Random Sampling: Selecting individuals randomly from a population to be in the sample. | Random Assignment: Assigning participants to different treatment groups randomly in an experiment.

T-distribution: Used when the population standard deviation is unknown and estimated by the sample standard deviation, heavier tails. | Normal Distribution: Used when the population standard deviation is known or with large sample sizes due to CLT, lighter tails.

Type I error: Rejecting the null hypothesis when it is true (false positive). | Type II error: Failing to reject the null hypothesis when it is false (false negative).

Standard deviation: Measures the spread of data in a single sample. | Standard error: Measures the variability of the sample statistic (e.g., sample mean) across multiple samples.

Confidence Intervals: Estimate a population parameter with a range of plausible values. | Hypothesis Tests: Assess the evidence against a specific claim about a population parameter.