chapter9 - Introduction to Probability and Statistics...

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1 Introduction to Probability and Statistics Thirteenth Edition Chapter 9 Large-Sample Tests of Hypotheses Introduction • Suppose that a pharmaceutical company is concerned that the mean potency m of an antibiotic meet the minimum government potency standards. They need to decide between two possibilities: The mean potency m does not exceed the mean allowable potency. The mean potency m exceeds the mean allowable potency. •This is an example of a test of hypothesis. Introduction • Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities: The person is guilty. The person is innocent. • To begin with, the person is assumed innocent. • The prosecutor presents evidence, trying to convince the jury to reject the original assumption of innocence, and conclude that the person is guilty. Parts of a Statistical Test 1. The null hypothesis, H 0 : Assumed to be true until we can prove otherwise. 2. The alternative hypothesis, H a : Will be accepted as true if we can disprove H 0 Court trial: Pharmaceuticals: H 0 : innocent H 0 : m does not exceeds allowed amount H a : guilty H a : m exceeds allowed amount Parts of a Statistical Test 3. The test statistic and its p -value: A single statistic calculated from the sample which will allow us to reject or not reject H 0 , and A probability, calculated from the test statistic that measures whether the test statistic is likely or unlikely , assuming H 0 is true. 4. The rejection region: A rule that tells us for which values of the test statistic, or for which p -values, the null hypothesis should be rejected. Parts of a Statistical Test 5. Conclusion: Either ―Reject H 0 ‖ or ―Do not reject H 0 ‖, along with a statement about the reliability of your conclusion. How do you decide when to reject H 0 ? Depends on the significance level, a, the maximum tolerable risk you want to have of making a mistake, if you decide to reject H 0 . Usually, the significance level is a = .01 or a = .05.
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2 Example The mayor of a small city claims that the average income in his city is $35,000 with a standard deviation of $5000. We take a sample of 64 families, and find that their average income is $30,000. Is his claim correct? 1-2. We want to test the hypothesis : H 0 : m = 35,000 (mayor is correct) versus H a : m  35,000 (mayor is wrong) Start by assuming that H 0 is true and m = 35,000. Example 3. The best estimate of the population mean m is the sample mean, $30,000 : From the Central Limit Theorem the sample mean has an approximate normal distribution with mean m = 35,000 and standard error SE = 5000/8 = 625. The sample mean, $30,000 lies z = (30,000 – 35,000)/625 = -8 standard deviations below the mean. The probability of observing a sample mean this far from m = 35,000 (assuming H 0 is true) is nearly zero .
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chapter9 - Introduction to Probability and Statistics...

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