How big the interval should be we will use a

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: interval. How big the interval should be? We will use a probabilistic way of doing this. Confidence Interval General Form Suppose unknown µ (or any population parameter) is of interest. And we figured out that there are two values, L and U such that when we calculate the probability P (L < µ < U ) = 1 − α where α is in (0, 1). Then the interval (L, U ) is called the 100(1 − α)% confidence interval. Is this a good idea? We will see. . . . But for now, let’s try to do what this asks for. Utku Suleymanoglu (UMich) Interval Estimation 3 / 17 Interval Estimation of Population Mean Case 1: Normally distributed population, σ known We learned that if X ∼ N (µ, σ 2 ) then X ∼ N (µ, σ 2 /n). ¯ X −µ So that Z = σ/√n has a standard normal distribution. Then we know that there is a value zα/2 such that 1 − α = P (−zα/2 < Z < zα/2 ) X −µ √ < zα/2 ) σ/ n √ √ = P − zα/2 (σ/ n) < X − µ < zα/2 (σ/ n) = P (−zα/2 < √ √ = P − X − zα/2 (σ/ n) < −µ < −X + zα/2 (σ/ n) √ √ = P X − zα/2 (σ/ n) < µ < X + zα/2 (σ/ n) =P L<µ<U √ √ where L = X...
View Full Document

This note was uploaded on 03/17/2014 for the course ECON 404 taught by Professor Staff during the Spring '08 term at University of Michigan.

Ask a homework question - tutors are online