Lecture16standard - Statistics 511: Statistical Methods Dr....

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Unformatted text preview: Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Lecture 15: Tests about Population Means and Population Proportions Devore: Section 8.2-8.3 April, 2011 Page 1 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 A Normal Population with known This case is not common in practice. We will use it to illustrate basic principles of test procedure design Let X 1 ,...,X n be a sample size n from the normal population. The null value of the mean is usually denoted and we consider testing either of the three possible alternatives > , < and 6 = The test statistic that we will use is Z = X- / n It measures the distance of X from in standard deviation units. April, 2011 Page 2 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Consider H a : > as an alternative. The outcome that would allow us to reject the null hypothesis H : = is z c for some c > How do you select c ? We need to control the probability of Type I Error. For a test of level , we have = P ( Type I Error ) = P ( Z c | Z N (0 , 1)) Therefore, we need to choose c = z . Such a test procedure is called upper-tailed . It is easy to understand that for H a : < we will have the rejection region of the form z c . For the test to have the level , we need to choose c =- z . Such a test is called a lower-tailed test . April, 2011 Page 3 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Now consider the case of H a : 6 = . The rejection region here consists of z c and z - c . For simplicity, consider the case = 0 . 05 . Then, . 05 = P ( Z c or Z - c | Z N (0 , 1)) = (- c ) + 1- ( c ) = 2[1- ( c )] Therefore, we select c such that 1- ( c ) = P ( Z c ) = 0 . 025 ; it is z . 025 = 1 . 96 . This test is called a two-tailed test . April, 2011 Page 4 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Summary Let H : = ; define the test statistic Z = X- / n . 1. H a : > has the rejection region z z and is called an upper-tailed test 2. H a : < has the rejection region z - z and is called an lower-tailed test 3. H a : 6 = has the rejection region z z / 2 or z - z / 2 and is called a two-tailed test April, 2011 Page 5 Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Recommended Steps for Testing Hypotheses about a Parameter 1. Identify the parameter of interest and describe it in the context of the problem situation. 2. Determine the null value and state the null hypothesis....
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This note was uploaded on 02/20/2012 for the course STAT 511 taught by Professor Bud during the Fall '08 term at Purdue University-West Lafayette.

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Lecture16standard - Statistics 511: Statistical Methods Dr....

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