Ch8 Hypothesis Testing

Ch8 Hypothesis Testing - Stat1801 Probability and...

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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Chapter VIII Hypothesis Testing § 8.1 Introduction Hypothesis A statistical hypothesis is an assertion or statement about the population, usually formulated in terms of population parameters. It is denoted by or . 0 H 1 H Null Hypothesis The null hypothesis , denoted by , is usually a statement about something that 0 H has been established, or something that has an authoritative standing, or something worth protecting. Alternative Hypothesis The alternative hypothesis , denoted by , is usually a statement about something 1 H that challenges the authority, or something that needs not be protected strongly. The hypotheses are often expressed in terms of the population parameters. However, there exists exceptional cases in non-parametric problems. Example 8.1 = 118 : 118 : 1 0 μ H H , , = 1 : 1 : 2 1 2 0 σ H H = 2 1 1 2 1 0 2 : 2 : p p H p p H on. distributi normal not is Population : on. distributi normal is Population : 1 0 H H Definition A test of statistical hypothesis is a procedure, based upon the observed values of the random sample obtained, that leads to the rejection or non-rejection of the null hypothesis . 0 H P.165
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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Example 8.2 An intensive survey conducted a few years ago found that the mean IQ of HKU students was 118. Some people suspect that the mean IQ has decreased recently. Which claim should we take? Null hypothesis: 118 : 0 = μ H Alternative hypothesis: 118 : 1 < H We may draw a random sample of HKU students and estimate the population mean IQ by the sample mean X . A reasonable test would be the procedure “Reject if 0 H 113 < X .” Note that X is just a point estimator of and there would be some estimation error. It may be still possible that the population mean IQ is really 118 but the observed sample mean X is smaller than 113, thereby leading to an incorrect conclusion by rejecting . To assess the reliability of the test, there are two possible types of error to be considered. 0 H Two types of errors Accept 0 H Reject 0 H 0 H true Correct decision Type I error ( false) true 0 H 1 H Type II error Correct decision Type I error probability: ( ) true | Reject 0 0 H H P = α Type II error probability: ( ) true | Accept 1 0 H H P = β There is a trade-off between the two types of error. Making smaller will lead to a larger , and vice versa. Therefore in designing a test we can only control one of them, and the convention is to guarantee in a desired low level and then try to reduce as much as we could (i.e. type I error is considered as more serious than type II error). That is why the roles of and are not interchangeable. 0 H 1 H In testing statistical hypothesis, we are testing against , i.e. we observe the data to see if there is sufficient evidence to reject . If we have sufficient 0 H 1 H 0 H P.166
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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 evidence to reject , we can have great confidence that is false and is true. However, if we observe the data and find that
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This note was uploaded on 02/01/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.

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Ch8 Hypothesis Testing - Stat1801 Probability and...

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