OHCh5-B - Chapter 5 Con dence Interval Estimation and...

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Chapter 5: Con°dence Interval Estimation and Hypothesis Testing Lecture 2
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1 HYPOTHESIS TESTING 1.1 Test Statistics We °rst hypothesize that the true parameter takes a particular numerical value, e.g., ° 1 = 0 in a regression with K = 1. Our task now is to "test" this hypothesis. The null hypothesis H 0 : ° 1 = 0 : (1)
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For this null hypothesis, three possible forms of an alternative hypothesis H A : ° 1 6 = 0 ; (2) which is used for a two-sided text . H A : ° 1 > 0 ; (3) which is used for a one-sided test . H A : ° 1 < 0 ; (4)
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which is also used for a one-sided test. Example 1: Consider the model- HS = ° 0 + ° 1 GDP , where HS denotes Housing Starts. A simple economic theory might claim that ° 1 > 0, and we might be interested in testing this theory. The alternative hypothesis is then given by Equation (3). Sometimes, the null hypothesis may not take the form that the parameter is zero. Then the alternative hypothesis changes accordingly. Example 2: Suppose a pharmaceuticals company claims that they have discovered a drug with a higher curing rate, r (measured in percentage), than the existing drugs which cure at a rate of 40%. Then the null hy- pothesis is
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H 0 : r = 40 (5) The alternative hypothesis is written as H 1 : r > 40 : (6) 1.2 Test Statistics A test statistic is a random variable we use to test the null hypothesis against an alternative hypothesis.
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t = ^ ° 1 ° ° 1 SE ( ^ ° 1 ) ; (7) follows the t distribution with ( N ° 2) degrees of freedom. Compute the t value under H 0 : ° 1 = 0: t = ^ ° 1 SE ( ^ ° 1 ) ; (8) follows the t distribution with ( N ° 2) degrees of freedom when the null hypothesis is true. This test statistic is called the t - statistic , and the test based on this test statistic is called the t - test .
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If H 0 is true, we expect a small number for j t j . While if H 0 is not true, we expect a large number for j t j . It makes sense to choose a number t C , and
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