lecture 14 Student's t

lecture 14 Student's t - The Problem with z Tests •...

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Unformatted text preview: The Problem with z Tests • Untenability of the known-variance 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 /( ν-2) • Symmetrical and bell-shaped 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 chi-square) E( χ 2 ( ν ) ) = ν , Var( χ 2 ( ν ) ) = 2 ν s 2 (n-1)/ 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.

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lecture 14 Student's t - The Problem with z Tests •...

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