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Unformatted text preview: Chapter 8. Tests of Hypotheses based on a Single Sample When the objective of an investigation is to estimate parameters, point estimate or confidence interval When the objective is to decide which of two contradictory claims about the parameter is correct Hypotheses Testing 8.1 Hypotheses and Test Procedures (statistical) hypothesis – a claim or an assertion either About the value of a single parameter About the values of several parameters About the form of an entire probability distribution Eg) 0. 75 μ = , 0. 10 p < , 1 2 μ μ = Def) Hypothesis Null hypothesis ( H ) – the claim that is initially assumed to be true (prior belief) Alternative hypothesis ( a H ) – the assertion that is contradictory to H Decision – reject H or fail to (do not ) reject H 1 Reject H in favor of a H only if sample evidence suggests that H is false, Otherwise, we continue to believe in the truth of H Eg) A company considers a new type of coating on bearings that it produces. The average wear life of current coating – 1000 hrs μ true average life of the new coating H : μ =1000, a H : μ >1000 Eg) Scientific research tries to decide whether a current theory ( H ) should be replaced by a more plausible and satisfactory explanation ( a H ) of the phenomenon under investigation p = proportion of defective board resulting from the changed process the current proportion of defective board = 0.1 H : p=0.1, a H : p<0.1 Null hypothesis will always be stated as an equality claim H : μ =1000 (or μ ≤ 1000), a H : μ >1000 H : p=0.1 (or p ≥ 0.1), a H : p<0.1 2 θ : parameter of interest, θ : null value of the parameter : H θ θ = : a H θ θ , : a H θ θ < , : a H θ θ ≠ Alternative Hypothesis: researcher’s hypothesis the claim that the researcher would really like to validate Test Procedures : specify 1. test statistic – a function of the sample data on which the decision is based 2. rejection (critical) region – the set of all test statistic values for which the null hypothesis will be rejected Reject H if test statistic value falls in the rejection region Def) Errors in hypotheses testing Type I error – reject H when it is true Type II error – do not rejecting H when it is false nature H is true H is false ( a H is true) decision Reject H Type I( α ) good Do not reject H good Type II( β ) 3 4 Ex 8.1) no visible damage 25% of the time in 10mph crash A new bumper is proposed to increase this percentage p – proportion of no visible damage of new bumper : 0. 25 : 0. 25 a H p vs H p = Experiment with n=20 independent crashes and set X = # of crashes with no visible damages R = {x: x ≥ 8} Reject : 0. 25 H p = if x...
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 Fall '08
 BUD
 Statistics, Null hypothesis, Statistical hypothesis testing

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