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# How big the interval should be we will use a

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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...
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## 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.

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