8.1-8.2 - Section 8.1, 8.2, and 8.4 Testing Statistical...

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Section 8.1, 8.2, and 8.4 Testing Statistical Hypotheses (TSH) 1. Basic Concepts of TSH Estimation deals with obtaining a plausible estimate of θ or obtaining a plausible interval (random) which contains . TSH deals with how to make a decision between two competing claims (hypotheses). Definition : Statistical hypothesis or simply a hypothesis is an assertion about one or more population characteristics or about the form of population distribution. Examples: (i) Let p = proportion of students having cases Hypothesis: .6 p . (ii) Let X = height of the students Hypothesis: 5) (170, ~ N X Kinds of hypotheses: Null hypothesis 0 H is a hypothesis which is initially assumed to be true. Alternative hypothesis a H is the hypothesis compared against 0 H . 1
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A test of hypothesis is a method or procedure of deciding between 0 H and a H , based on the sample information. If sample contains strong evidence against 0 H , then it will be rejected in favor of a H . Otherwise, we will continue to believe in 0 H . Hypothesis always refer to some population or model, not to a particular outcome. For this reason, we must state H 0 and H a in terms of population parameters. Example : Suppose 10) , ( ~ μ N X , consider testing (i) 50 : 0 = H against 50 : 1 < H (lower-tailed) or 50 : a H (upper-tailed) (ii) 50 : 0 = H against 50 : a H (two-tailed). A test leads to one of the following decisions (i) Fail to reject 0 H (ii) Reject 0 H (Accept a H ) Obviously, we have two-kinds of errors inherent in any test. 2
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Definition: Type I error Reject 0 H when 0 H is true. Type II error Fail to reject 0 H when a H is true. Definition: P(Type I error) = α = P (Reject 0 H when 0 H is true). P(Type II error) = β = P(Fail to reject 0 H when a H is true). It can be shown that both errors and can not be minimized. Usually, we fix one of the errors, say , and choose a test that minimizes . In practice, = .05 or = 0.01, is chosen. Definition: The specified value of is also called “level of significance” or “significance level of the test”. Remarks : (i) A test with significance level = .05 means that when the test repeated several times, only 5% of times it may commit the type I error. (ii) The quantity (1- ) = P (Reject 0 H when a H is true) is called the power of the test .
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8.1-8.2 - Section 8.1, 8.2, and 8.4 Testing Statistical...

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