Section 8.1-8.2 - Section 8.1 8.2 Testing Statistical...

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Unformatted text preview: Section 8.1- 8.2 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 θ . 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 cars Hypothesis: .6 ≥ p . (ii) Let X = height of the students Hypothesis: 5) (170, ~ N X Kinds of hypotheses: Null hypothesis H is a hypothesis which is initially assumed to be true. Alternative hypothesis 1 H is the hypothesis compared against H . 1 A test of hypothesis is a method or procedure of deciding between H and 1 H , based on the sample information. If sample contains strong evidence against H , then it will be rejected in favor of 1 H . Otherwise, we will continue to believe in H . Example 1 Suppose 10) , ( ~ μ N X and consider testing (i) 50 : = μ H against 50 : 1 < μ H (lower-tailed) or 50 : 1 μ H (upper-tailed) (ii) 50 : = μ H against 50 : 1 ≠ μ H (two-tailed). A test leads to one of the following decisions (i) Accept H (ii) Reject H (Accept 1 H ) Obviously, we have two-kinds of errors are inherent in any test. Definition: Type I error ≡ Reject H when H is true. Type II error ≡ Accepting H when 1 H is true. Definition: P(Type I error) = α = P (Reject H when H is true). P(Type II error) = β = P(Accepting H when 1 H is true). 2 It can be shown that both errors α and β can not be minimized. Usually, we fix one of the errors, say α , and choose a test that minimizes β . In practice, α = .05 or α = 0.01, is chosen. Definition: The specified value of α is also called “ level of significance” or “ significance level of the test ”. Remarks : (i) A test with significance level α = .05 means that when the test repeated several times, only 5% of times it may commit the type I error. (ii) The quantity (1- β ) = P (Reject H when 1 H is true) is called the power of the test. (iii) If α decreases then β increases. Usually, the largest tolerable value of α is decided based on the problem under consideration. Then a test is found with P (Type I error) α ≤ and β is minimum or (1- β ) = power is maximum. Test Statistics and P – values The test is carried out using a test statistic, usually a standardized function of the data. 3 Example 1 : Let μ ≡ population mean. Suppose we want to test 50 : = μ H against 50 : 1 < μ H . Obviously, sample mean x provides information about μ . Also, we know n x σ μ , N ~ When σ is not known and n is large, (0,1) N ) ( 2245- s n x μ distribution....
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This note was uploaded on 07/25/2008 for the course STT 351 taught by Professor Palaniappan during the Summer '08 term at Michigan State University.

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Section 8.1-8.2 - Section 8.1 8.2 Testing Statistical...

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