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# Lecture16standard - Statistics 511 Statistical Methods Dr...

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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

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 A Normal Population with known σ 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 μ 0 and we consider testing either of the three possible alternatives μ > μ 0 , μ < μ 0 and μ 6 = μ 0 The test statistic that we will use is Z = ¯ X - μ 0 σ/ n It measures the distance of ¯ X from μ 0 in standard deviation units. April, 2011 Page 2
Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Consider H a : μ > μ 0 as an alternative. The outcome that would allow us to reject the null hypothesis H 0 : μ = μ 0 is z c for some c > 0 How do you select c ? We need to control the probability of Type I Error. For a test of level α , we have α = P ( Type I Error ) = P ( Z c | Z N (0 , 1)) Therefore, we need to choose c = z α . Such a test procedure is called upper-tailed . It is easy to understand that for H a : μ < μ 0 we will have the rejection region of the form z c . For the test to have the level α , we need to choose c = - z α . Such a test is called a lower-tailed test . April, 2011 Page 3

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Statistics 511: Statistical Methods Dr. Levine Purdue University Spring 2011 Now consider the case of H a : μ 6 = μ 0 . The rejection region here consists of z c and z ≤ - c . For simplicity, consider the case α = 0 . 05 . Then, 0 . 05 = P ( Z c or Z ≤ - c | Z N (0 , 1)) = Φ( - c ) + 1 - Φ( c ) = 2[1 - Φ( c )] Therefore, we select c such that 1 - Φ( c ) = P ( Z c ) = 0 . 025 ; it is z 0 . 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 0 : μ = μ 0 ; define the test statistic Z = ¯ X - μ 0 σ/ n . 1. H a : μ > μ 0 has the rejection region z z α and is called an upper-tailed test 2. H a : μ < μ 0 has the rejection region z ≤ - z α and is called an lower-tailed test 3. H a : μ 6 = μ 0 has the rejection region z z α/ 2 or z ≤ - z α/ 2 and is called a two-tailed test April, 2011 Page 5

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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. 3. State the alternative hypothesis. 4. Give the formula for the computed value of the test statistic. 5. State the rejection region for the selected significance level 6. Compute any necessary sample quantities, substitute into the formula for the test statistic value, and compute that value.
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