M312_4 - Probability and Statistics II M-312 Lecture 4 4...

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Unformatted text preview: Probability and Statistics II M-312 Lecture 4 4. Tests of Hypotheses Based on a Single Sample 4.1. Hypotheses and Test Procedures Definition 4.1. The null hypothesis , denoted by H , is the claim that is initially assumed to be true (the prior belief claim). The alternative hypothesis , denoted by H a , is the assertion that is contradictory to H . The null hypothesis will be rejected in favor to the alternative hypothesis only if sample evidence suggests that H is false. If the sample does not strongly contradict H , we will continue to believe in the truth of the null hypothesis. The two possible conclusions from a hypothesis-testing analysis are then reject H or fail to reject H . TEST PROCEDURES Definition 4.2. A type I error consists of rejecting the null hypothesis H when it is true. A type II error involves not rejecting H when H is false. Example 4.1. A certain type of automobile is known to sustain no visible damage 25% of the time in 10-mph crash tests. A modified bumper design has been proposed in an effort to increase this percentage. Let p denote the proportion of all 10-mph crashes with this new bumper that results in no visible damage. The hypotheses to be tested are H : p = . 25 (no improvement) versus H a : p > .25. The test will be based on an experiment involving n = 20 independent crashes with prototypes of the new design. Intuitively, H 0 should be rejected if a substantial number of the crashes show no damage. Consider the following test procedure: Test statistic: X = the number of crashes with no visible damage. Rejection region: R 8 = {8, 9, 10,, 19, 20}; that is, reject H if , 8 x where x is the observed value of the test statistic. 1. A test statistic , a function of the sample data on which the decision (reject H 0 or do not reject H ) is to be based 2. A rejection region , the set of all test statistic values for which H will be rejected The null hypothesis will then be rejected if and only if the observed or computed test statistic value falls in the rejection region. 2 This rejection region is called upper-tailed because it consists only of large values of the test statistic. When H is true, X a binomial probability distribution with n = 20 and p = .25. Then = P (type I error) = P ( H is rejected when it is true) = P ( X 8 when X ~ Bin (20, .25) ) = 1 B (7; 20, .25) = 1 - .898 = .102 That is, when H is actually true, roughly 10% of all experiments consisting of 20 crashes would result in H being incorrectly rejected (a type I error). In contrast to , there is not a single . Instead, there is a different for each different p that exceeds .25. Thus there is a value of for p = .3 (in which case X ~ Bin (20, .3)), another value of for p = .5, and so on. For example, (.3) = P (type II error when p = .3) = P ( H is not rejected when it is false because p = .3) = P ( X 7 when X ~ Bin (20, .3) ) = B (7; 20, .3) = .772 When p is actually .3 rather than .25 (a small departure from...
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M312_4 - Probability and Statistics II M-312 Lecture 4 4...

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