Lesson 19: T-Distribution

Student's t-distribution

A probability distribution used to estimate population parameters when the sample size is small (n < 30) and/or the population variance is unknown.

Developer of the t-distribution

William Sealy Gosset in 1908.

Origin of the name "Student"

Gosset used "Student" as a pen name or pseudonym when publishing his paper describing the distribution.

Gosset's background

He worked at a brewery and was interested in small sample problems, such as the chemical properties of barley.

Formula for the t-statistic

t = (x̄ - μ) / (s / √(n)), where s is the sample standard deviation.

Degrees of freedom formula

df = n - 1, which is one less than the sample size.

Degrees of freedom example

If the sample size n is 25, the number of degrees of freedom is 24.

Symmetry of the t-distribution

It is symmetrical about 0, meaning a segment from the peak to 0 divides the curve into two equal areas.

Shape of the t-distribution

Bell-shaped like the normal distribution, but with heavier tails and greater variance due to estimating the population standard deviation with s.

Mean, median, and mode of the t-distribution

All are equal to zero.

Variance of the t-distribution

Always greater than 1, calculated as v / (v - 2) where v is the number of degrees of freedom.

Effect of increasing degrees of freedom

As degrees of freedom increase, the t-distribution curve looks more and more like the standard normal distribution.

Standard deviation of the t-distribution

It varies with the sample size and is always greater than 1.

Total area under a t-distribution curve

Equal to 1 or 100%, representing probabilities associated with specific sets of t-values.

Z-score formula

z = (x̄ - μ) / (σ / √(n)), where x̄ is sample mean, μ is population mean, σ is population standard deviation, and n is sample size.

Standardizing a single observation

If standardizing a single observation, the value of n is 1, and the formula becomes z = (x - μ) / σ.

One-tailed test directions

Can be right-tailed (positive) or left-tailed (negative).

Two-tailed test notation

Represented by ± to account for both tails of the distribution.