Confidence intervals

Population mean Assumption: sample from normal distribution

1: Population variance  2 is known

Given X 1, X 2, …, X N – independent, normally distributed, what is the distribution of ?

Confidence interval 95% of data

With probability 0.95

Equivalently

2: Population variance  2 is not known Use estimate:

Not true. Intuition tells us that the interval should be wider

E.G., N=7 95% CI looks like this

Student t distribution With N degrees of freedom (d.f.)

Gamma function L. Euler, XXVIII century

Normal samples and t distribution X 1, X 2, …X N – independent normal Has t distribution with N-1 d.f.

Linear regression and t - distribution U – normally distributed disturbance x i - controlled variable, fixed values

Assumption

Leads to

Residual mean square – estimates variance of normal disturbance U

Follow t distribution with n-2 degrees of freedom

Non normal samples For large n, estimates can still be used due to Central Limit Thorem