1. Confidence intervals

2. Population mean Assumption: sample from normal distribution

3. 1: Population variance  2 is known

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

5. Confidence interval 95% of data

6. With probability 0.95

7. Equivalently

8. 2: Population variance  2 is not known Use estimate:

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

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

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

12. Gamma function L. Euler, XXVIII century

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

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

16. Assumption

17. Leads to

18. Residual mean square – estimates variance of normal disturbance U

19. Follow t distribution with n-2 degrees of freedom

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