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Chapter 8

Chapter 8 - Chapter 8 Tests of Hypotheses based on a Single...

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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 10-mph 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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Chapter 8 - Chapter 8 Tests of Hypotheses based on a Single...

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