Random sampling ensures that the sample is representative of the population, reducing bias and allowing for valid inferences.

Random assignment ensures that any observed differences between treatment groups are due to the treatment itself, rather than confounding variables.

If we were to take many samples and construct 95% confidence intervals, about 95% of those intervals would contain the true difference in population means.

Increasing the sample size decreases the width of the confidence interval, providing a more precise estimate of the population parameter.

It suggests that there is no significant difference between the means of the two populations, as zero is a plausible value for the difference.

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

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

Point Estimate ± Margin of Error

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

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

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.