X so that z n has a standard normal distribution then

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Unformatted text preview: hat 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 − zα/2 σ/ n and U = X + zα/2 σ/ n Utku Suleymanoglu (UMich) Interval Estimation 4 / 16 Interval Estimation of Population Mean We figured out the lower and upper bounds of the 100(1 − α)% confidence interval. Notice that they are in the format (as promised) x ± ME . ¯ √ Margin of error in this case is zα/2 σ/ n. Or zα/2 σx ¯ Let’s summarize this result before we discuss what it means: CI for µ, Case 1: Normal Population, σ known If population has a normal distribution and σ is known, a 100(1 − α)% confidence interval for µ can be constructed via: √ x ± zα/2 σ/ n ¯ where zα/2 is the z value such that F (z ) = 1 − α/2 or simply z value with upper tail probability of α/2. Now let’s do an example. Utku Suleymanoglu (UMich) Interval Estimation 5 / 16 Interval Estimation of Population Mean Example Suppose you have a normally distributed population with variance σ 2 = 4. You have a sample of 400 observations, whose sample average is x = 3. Let’s build a 90% confidence ¯ interval for µ. If (1 − α) = 0.90, then α = 0.10 and α/2 = 0.05. What is th...
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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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