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# U-M ECON 405 - ECON 405 LECTURE NOTES

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TodaySmall Sample Hypothesis Test for the Population MeanSmall Sample Hypothesis Test for the Population Mean (cont.)Example:PowerPoint PresentationRelationships Between Tests and CI’sExample (3.96)Slide 9Large Sample Inferences for ProportionsSlide 11Slide 12Slide 13Large Sample Hypothesis Test for the Population ProportionSlide 15Slide 16ExampleToday•Today: Chapter 10•Sections from Chapter 10: 10.1-10.4•Recommended Questions: 10.1, 10.2, 10-8, 10-10, 10.17, 10.19Small Sample Hypothesis Test for the Population Mean•Have a random sample of size n ; x1, x2, …, xn• •Test Statistic:00:HnSXt/Small Sample Hypothesis Test for the Population Mean (cont.)•P-value depends on the alternative hypothesis:– – – •Where T represents the t-distribution with (n-1 ) degrees of freedom)( value-p : :01tTPH )( value-p : :01tTPH |)| (2 value-p : :01tTPH Example:•An ice-cream company claims its product contains 500 calories per pint on average•To test this claim, 24 of the company’s one-pint containers were randomly selected and the calories per pint measured•The sample mean and standard deviation were found to be 507 and 21 calories•At the 0.01 level of significance, test the company’s claim•What assumptions do we make when using a t-test?•Can use t procedures even when population distribution is not normal. Why?Relationships Between Tests and CI’s•Confidence interval gives a plausible range of values for a population parameter based on the sample data•Hypothesis Test assesses whether data gives evidence that a hypothesized value of the population parameter is plausible or implausible•Seem to be doing something similar•For testing:•If the test rejects the null hypothesis, then•If the null hypothesis is not rejected,0100: vs.: HHExample (3.96)•Based on a random sample of size 18 from a normal population, an investigator computes a 95% confidence interval for the mean and gets [27.1, 39.3]•What is the conclusion of the t-test at the 5% level for:– – 29: vs.29:10HH8.26: vs.8.26:10HH•Suppose we reject the second null hypothesis at the 5% level•Another experimenter wishes to perform the test at the 10% level…would they reject the null hypothesis•Another experimenter wishes to perform the test at the 1% level…would they reject the null hypothesis•What does changing the significance level do to the range of values for which we would reject the null hypothesisLarge Sample Inferences for ProportionsExample:•Consider 2 court cases:–Company hires 40 women in last 100 hires–Company hires 400 women in last 1000 hires•Is there evidence of discrimination?•Can view hiring process as a Bernoulli distribution:•Want to test:Situation:•Want to estimate the population proportion (probability of a “success”), p•Select a random sample of size n•Record number of successes, X•Estimate of the sample proportion is:•If n is large, what is distribution of•Can use this distribution to test hypotheses about proportions pˆLarge Sample Hypothesis Test for the Population Proportion•Have a random sample of size n• • •Test Statistic:00: ppH nqpppZ/ˆ000nXp ˆ•P-value depends on the alternative hypothesis:– – – •Where Z represents the standard normal distribution)( value-p : :01zZPppH )( value-p : :01zZPppH |)| (2 value-p : :01zZPppH •What assumptions must we make when doing large sample hypotheses tests about proportions?•Example revisited:Example•For both court cases, find a 95% confidence interval for the probability that the company hires a

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