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Unformatted text preview: Confidence intervals Suppose we have a set of n measurements and our model for them is that they are inde- pendent and distributed like X where E ( X ) = μ and SD( X ) = σ . Here μ and σ describe the “state of the world” (as relevant to our experiment) but are not known. In addition, while we don’t know what is causing the variation in measurements, we don’t believe any important factor is changing between measurements so μ and σ 2 remain constant during the sampling and measurement process. Confidence intervals for μ Given some data we want to determine a range of possible μ values that are “reasonable” in a sense we will now consider. We bring together some of the things we saw earlier in the course: (i) we know from the Central Limit Theorem that the sample mean random variable X is approximately N ( μ, σ 2 /n ) (ii) for any number α between 0 and 1 we can use standard Normal tables to find the value z α > 0 such that Φ( z α )- Φ(- z α ) = 1- α ; (iii) hence P- z α < X- μ σ/ √ n < z α = 1- α or P X- μ σ/ √ n > z α = P X- μ > z α σ √ n = α In words this means whatever the true value of μ is, the chance that we observe a sample mean...
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This note was uploaded on 05/12/2010 for the course APPLIED ST 2010 taught by Professor Various during the Spring '10 term at Universidad Nacional Agraria La Molina.
- Spring '10