$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $t^*$ is the critical t-value, s is the sample standard deviation, and n is the sample size.

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

$Margin of Error = t^* \times \frac{s}{\sqrt{n}}$

Point Estimate $\pm$ Margin of Error

$SE = \frac{s}{\sqrt{n}}$, where s is the sample standard deviation and n is the sample size.

t-distribution: Used when σ is unknown, heavier tails, shape depends on degrees of freedom. | Normal distribution: Used when σ is known, lighter tails, defined by mean and standard deviation.

t-test: Used when population standard deviation is unknown, uses t-distribution, appropriate for small sample sizes. | z-test: Used when population standard deviation is known, uses normal distribution, appropriate for large sample sizes.

Increasing sample size: Decreases width of confidence interval, reduces margin of error. | Increasing confidence level: Increases width of confidence interval, increases margin of error.

Sample mean: Calculated from a subset of the population, used to estimate population mean, varies from sample to sample. | Population mean: True average of the entire population, usually unknown, constant value.

Standard deviation: Measures the spread of individual data points around the sample mean. | Standard error: Measures the spread of sample means around the population mean, estimates the variability of the sample mean.

A probability distribution used for estimating population means when the sample size is small and the population variance is unknown. It has heavier tails than the normal distribution.

The number of independent pieces of information available to estimate a parameter. For a one-sample t-test, df = n - 1, where n is the sample size.

A single value estimate for a population parameter. For estimating a population mean, the sample mean (x̄) is used as the point estimate.

The range of values above and below the sample statistic in a confidence interval. It quantifies the uncertainty in estimating the population parameter.

A range of values calculated from sample data that is likely to contain the true population parameter with a certain level of confidence.