Point estimate ± (Critical value) * (Standard error)

df = n - 2, where n is the sample size.

$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

$\chi^2 = \sum \frac{(Observed - Expected)^2}{Expected}$

A t-test used to compare the means of two related groups, such as before-and-after measurements on the same subjects. This reduces variability by analyzing the differences within each pair.

The number of independent pieces of information available to estimate a parameter. It affects the shape of the t-distribution and chi-square distribution.

The probability of rejecting the null hypothesis when it is true (Type I error). Commonly set at 0.05, meaning there is a 5% risk of incorrectly rejecting the null.

A test to determine whether observed sample data matches a hypothesized distribution. It assesses if the observed frequencies significantly differ from expected frequencies.

Rejecting the null hypothesis when it is actually true. The probability of making a Type I error is equal to the significance level (alpha).

One Sample T-Test: Compares the mean of a single sample to a known value. | Two Sample T-Test: Compares the means of two independent samples to determine if there is a significant difference.

Chi-Square Test for Independence: Tests for association between two categorical variables within a single population. | Chi-Square Test for Homogeneity: Tests if the distribution of a categorical variable is the same across multiple populations.

One Proportion Z-Test: Tests a claim about a single population proportion. | Two Proportion Z-Test: Compares the proportions of two independent populations.

T-tests: Used when the population standard deviation is unknown and estimated from the sample. | Z-tests: Used when the population standard deviation is known.

Confidence Interval: Estimates a population parameter with a range of plausible values. | Hypothesis Test: Assesses evidence against a specific claim about a population parameter.