Section 8.1-8.2

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

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 18

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

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online