Lecture16standard

Lecture16standard - Statistics 511 Statistical Methods Dr...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of 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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of 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
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern