1. MATH408: Probability & Statistics Summer 1999 WEEKS 8 & 9 Dr. Srinivas R. Chakravarthy Professor of Mathematics and Statistics Kettering University (GMI Engineering & Management Institute) Flint, MI 48504-4898 Phone: 810.762.7906 Email: schakrav@kettering.edu Homepage: www.kettering.edu/~schakrav

2. ANALYSIS OF VARIANCE We saw how to study one population and two populations. What if we have to compare more than two? –Suppose that the production capacity of three shifts are to be compared. –Are the tensile strengths of three types of beams the same?

3. EXAMPLE

5. Linear statistical model

7. Total sum of squares

15. EXAMPLES

16. Linear Regression Analysis Analysis of relationships among variables. Evaluate the importance of individual predictor variables. Learn as much as possible about the environment. Mechanism for assessing and explaining variability inherent in the system. Regression analysis is one of the most widely used statistical techniques.

17. Statistical technique for investigating and modeling the relationship between the response (dependent) and explanatory (independent or predictor) variables. Applications of regression: almost every field, including medicine, economics, business, engineering and management. It utilizes the experimental data on the pertinent variables to see whether any relationship between the independent variables and the dependent variable of the model under study

18. In general regression methods are used to study the relationship, if any, and to predict the future values. Regression may be applied to correlating data in a wide range of problems ranging from the simple correlation of physical properties to the analysis of complex system. In many areas that require calibration study, regression analysis is frequently used. This is very important as the actual measurements may require very costly equipment, which may not be easily transportable.

19. In these cases, to minimize the cost and time involved in getting the measurements an alternative approach (cheaper and easy to implement) will be available. The form of the relationship (such as linear, quadratic, exponential) between explanatory and response variables may be guessed if something is known about the model. Frequently, a linear function is assumed initially if nothing is known about the model.

20. In almost all applications of regression, the fitted regression equation is only an approximation to the true relationship between variables. The fitted equation is very sensitive to the data Care should be taken to accurate data collection. Many of the techniques used in regression analysis can be distorted by false data.

21. Regression analysis and quality (a) process for building a useful analog of a production system (b) mechanism to aid in understanding complex interrelationship (c) mechanism for assessing and explaining variability inherent in the system (d) tool to focus on important phenomena affecting the system

22. Stages of a Comprehensive Regression analysis Stage 1: Investigate the data base. Plot the variables. Stage 2: Propose a functional form for each variable. Stage 3: Estimate the parameters of the model. Stage 4: Assess the model assumptions. Stage 5: Select the key regressor variables.

23. EXAMPLE

26. Basic Concepts Simple: There is one only regressor variable. Linear: The power of the parameters is at most 1. Multiple: More than one regressor variables. Order: The highest power of a regressor variable.

27. MULTIPLE LINEAR REGRESSION MODEL ( No interaction)

28. MULTIPLE LINEAR REGRESSION MODEL (with interaction)

29. MULTIPLE LINEAR REGRESSION MODEL ( Full Second-order Model)

42. HOMEWORK PROBLEMS Sections 6.1-6.4 1, 4, 5, 8, 10,12, 14