T-distribution

When a sufficiently large sample of n independent observations is taken from a population with mean μ and standard deviation σ, the sampling distribution of the sample mean x̄ will be nearly normal with mean x̄ = μ and standard error SE = σ/sqrt(n).

Central Limit Theorem

A distribution used to model the sample mean x̄ when conditions for the Central Limit Theorem are not met, providing thicker tails to account for the uncertainty when using the sample standard deviation s.

T-Distribution

We require independence of observations, the normality assumption, and that data come from a normally distributed population for the sampling distribution of x̄ to be nearly normal.

Conditions for CLT

The t-distribution has a bell shape, shallower than the normal distribution with thicker tails, centered at 0, and the degrees of freedom (df = n - 1) determine the shape.

Properties of T-Distribution

Calculated as x̄ ± t(df) * SE, where t* is based on the confidence level and t-distribution, to construct an interval for the sample mean x̄.

Confidence Intervals

Involves calculating the T-score as T = (x̄ - μ)/SE, finding the p-value, and comparing it against the significance level to draw conclusions.

Hypothesis Tests with T-Score

Define regions of extreme values of the test statistic under the null hypothesis, where if the observed test statistic falls within the rejection region, the null hypothesis is rejected; otherwise, it is not rejected.

Rejection Regions