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Unformatted text preview: The Problem with z Tests • Untenability of the knownvariance assumption • It is usually necessary to estimate the population variance using sample data s 2 = σ 2 = SS n 1 ^ z = (X  μ )/( σ / n) Õ t = (X  μ )/(s/ n) Õ 1 Student’s t • t ( ν ) the central variant of Student’s t • A family of distributions characterized by the single parameter, ν , with μ = 0 and σ 2 = ν /( ν2) • Symmetrical and bellshaped with fatter tails than the normal distribution • Asymptotically normal as ν approaches inFnity 2 Variables in the Test Statistic • The shape of the sampling distribution of t ( ν ) now depends on the sampling distribution of two variables, X , and s (used in place of the constant, σ ) • The dependence on the sampling distribution of s is responsible for the increased proportion of observations in the tails of the sampling distribution of t ( ν ) with smaller sample sizes 3 Sampling Distribution of the Variance • χ 2 ( ν ) (central chisquare) E( χ 2 ( ν ) ) = ν , Var( χ 2 ( ν ) ) = 2 ν s 2 (n1)/ σ 2 = ∑ (x  X) 2 / σ 2 ~ χ 2 (...
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This note was uploaded on 01/21/2009 for the course PSYC 60 taught by Professor Ard during the Winter '08 term at UCSD.
 Winter '08
 Ard

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