For a large enough sample size (n ≥ 30), the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.

Ensures independence when sampling without replacement; population size must be at least 10 times the sample size.

It means we don't have enough evidence to say the null hypothesis is false, not that it's true.

To challenge a claim about a population mean using sample data.

Use when the population standard deviation (σ) is unknown and you are working with means.

t-test: σ unknown, uses sample standard deviation (s), t-distribution | z-test: σ known, population normally distributed, uses z-distribution

Type I: Rejecting a true null hypothesis (false positive) | Type II: Failing to reject a false null hypothesis (false negative)

One-sample: Compares the mean of one sample to a known value. | Two-sample: Compares the means of two independent groups.

Confidence intervals: estimate a population parameter | Significance tests: test a claim about a population parameter

Standard deviation: measures the spread of individual data points in a sample. | Standard error: estimates the variability of sample means around the population mean.

$t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$ where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $s$ is the sample standard deviation, and $n$ is the sample size.

df = n - 1, where n is the sample size.

Point estimate ± (Critical value * Standard error)

$t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Critical Value * Standard Error