06StdNormalDist

# 06StdNormalDist - Statistical Techniques I EXST7005 The Z...

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Statistical Techniques I EXST7005 The Z distribution You are still here

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Objective The first statistical distribution we will use and develop for hypothesis testing is the Z distribution. This will require an understanding of the distribution, Of how to work with the distribution, And of the Z transformation. Fortunately, other statistical distributions will be similar. Once these techniques are learned, the apply readily to other statistical tests and applications.
THE Z - TRANSFORMATION Purpose - transforms values from any normal population to the corresponding values from the Standard Normal Distribution. this distribution is N( μ =0, σ =1) Zi = (Yi- μ ) / σ

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THE Z - TRANSFORMATION (continued) where; μ = the mean of the original population σ = the standard deviation of the original population Yi = the value of an observation from the original population Zi = the corresponding value from a Standard Normal Distribution
THE Z - TRANSFORMATION (continued) The purpose of this transformation is to deal wit the infinite number of possible normal curves with different values of μ and σ by standardizing any normal curve so we can work with a single distribution. we will then work with these distributions from a TABLE of Z values. This will tie together much of what we have discussed (frequency and probability concepts, transformations, use of means, variances and standard deviations, and their calculations).

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PROBABILITY statements "Typical" PROBABILITY statements are of the form. P[ Z Z0] = r.c.f. at Z0 where Z0 is an hypothesized value for Z < 0 Z > 0 0 0 Z 0 Z 0
PROBABILITY statements (continued) We will need tables to work with the Z distribution. You should have those tables available. Your book has Z-tables and I have a copy of mine on the internet. The Z table is exactly symmetric. As a result, th negative half (below zero) is a mirror image of the upper half. So our tables will give only half of the distribution, since it is EXACTLY symmetric.

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PROBABILITY statements (continued) To work with these half tables, it is important to note that P[Z 0] = P[Z 0] = 0.5 since half of the distribution is above 0 and half is below P[Z -Z0] = P[Z +Z0] since the table is symmetric P[Z Z0] = 1 - P[Z Z0]
TABULATED Z DISTRIBUTION My tables are patterned after Steel & Torrie, 1980, page 578 The table in the text is "ONE-SIDED". Only 1 side is required due to symmetry. Table gives positive values only.

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TABULATED Z DISTRIBUTION (continued) Values in the rows on the LEFT SIDE and TOP of the Z table give the value of Z, values in the body of the table are the probabilities of randomly choosing a larger Z value by random chance For example, take Z = 0.11. What proportion of the distribution occurs above this value? Or, what is the probability of picking a Z value at random and it being larger than 0.11?
0.00 0.01 0.02 0.03

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## This note was uploaded on 12/29/2011 for the course EXST 7087 taught by Professor Wang,j during the Fall '08 term at LSU.

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06StdNormalDist - Statistical Techniques I EXST7005 The Z...

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