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Correlation And Linear Regression

Correlation And Linear Regression. Chapter 7. Introduction Correlation And Linear Regression. Engineering Statistics. Correlation Analysis And Linear Regression. Chapter 7. Objectives:.

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Correlation And Linear Regression

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  1. Correlation And Linear Regression Chapter 7 Introduction Correlation And Linear Regression Engineering Statistics

  2. Correlation Analysis And Linear Regression Chapter 7 Objectives: In business and economic applications, frequently interest is in relationships between two or more random variables, and the association between variables is often approximated by postulating a linear functional form for their relationship. After working through this chapter, you should be able to: • understand the basic concepts of regression and correlation analyses; • determine both the nature and the strength of the linear relationship between two variables. Engineering Statistics 2

  3. Correlation Analysis And Linear Regression Chapter 7 Scatter Plots and Correlation Ascatter plot(or scatter diagram) is used to show the relationship between two variables Correlation analysis is used to measure strength of the association (linear relationship) between two variables Engineering Statistics 3

  4. Correlation Analysis And Linear Regression Chapter 7 Scatter Plot Examples Linear relationships Curvilinear relationships y y x x y y x x May 19 Engineering Statistics 4

  5. Correlation Analysis And Linear Regression Chapter 7 Scatter Plots and Correlation Strong relationships Weak relationships y y x x y y x x Engineering Statistics 5

  6. Correlation Analysis And Linear Regression Chapter 7 Scatter Plots and Correlation No relationship y x y x Engineering Statistics 6

  7. Correlation Analysis And Linear Regression Chapter 7 Correlation Coefficient The population correlation coefficient ρ (rho) measures the strength of the association between the variables The sample correlation coefficient r is an estimate of ρ and is used to measure the strength of the linear relationship in the sample observations May 19 Engineering Statistics 7

  8. Correlation Analysis And Linear Regression Chapter 7 Features of ρand r Unit free Range between -1 and 1 The closer to -1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker the linear relationship Engineering Statistics 8

  9. Correlation Analysis And Linear Regression Chapter 7 Examples of Approximate r Values y y y x x x r = -1 r = -.6 r = 0 y y x x r = +.3 r = +1 May 19 Engineering Statistics 9

  10. Correlation Analysis And Linear Regression Chapter 7 Calculating the Correlation Coefficient Sample correlation coefficient: or the algebraic equivalent: where: r = Sample correlation coefficient n = Sample size x = Value of the independent variable y = Value of the dependent variable May 19 Engineering Statistics 10

  11. Correlation Analysis And Linear Regression Chapter 7 Example 1 • Considering the following Data • Should the correlation be positive or negative • Calculate correlation May 19 Engineering Statistics 11

  12. Correlation Analysis And Linear Regression Chapter 7 Solution:- May 19 Engineering Statistics 12

  13. Correlation Analysis And Linear Regression Chapter 7 Example 2 The following are the number of minutes it took 10 mechanics to assemble a piece of machinery in the morning ,and in the late afternoon Calculate correlation Should the correlation be positive or negative Engineering Statistics 13

  14. Correlation Analysis And Linear Regression Chapter 7 Solution:- Engineering Statistics 14

  15. Correlation Analysis And Linear Regression Chapter 7 Solution:- Engineering Statistics 15

  16. Correlation Analysis And Linear Regression Chapter 7 Introduction to Regression Analysis • Regression analysis is used to: • Predict the value of a dependent variable based on the value of at least one independent variable • Explain the impact of changes in an independent variable on the dependent variable • Dependent variable: the variable we wish to explain • Independent variable: the variable used to explain the dependent variable Engineering Statistics 16

  17. Correlation Analysis And Linear Regression Chapter 7 Simple Linear Regression Model • Only one independent variable, x • Relationship between x and y is described by a linear function • Changes in y are assumed to be caused by changes in x Engineering Statistics 17

  18. Correlation Analysis And Linear Regression Chapter 7 Types of Regression Models Positive Linear Relationship Relationship NOT Linear Negative Linear Relationship No Relationship Engineering Statistics 18

  19. Correlation Analysis And Linear Regression Chapter 7 Population Linear Regression The population regression model: Random Error term, or residual Population SlopeCoefficient Population y intercept Independent Variable Dependent Variable Linear component Random Error component Engineering Statistics 19

  20. Correlation Analysis And Linear Regression Chapter 7 Linear Regression Assumptions • Error values (ε) are statistically independent • Error values are normally distributed for any given value of x • The probability distribution of the errors is normal • The probability distribution of the errors has constant variance • The underlying relationship between the x variable and the y variable is linear Engineering Statistics 20

  21. Correlation Analysis And Linear Regression Chapter 7 Population Linear Regression (continued) y Observed Value of y for xi εi Slope = β1 Predicted Value of y for xi Random Error for this x value Intercept = β0 x xi Engineering Statistics 21

  22. Correlation Analysis And Linear Regression Chapter 7 Determining the line of “best fit”: The sample regression line provides an estimate of the population regression line Estimated (or predicted) y value Estimate of the regression intercept Estimate of the regression slope Independent variable The individual random error terms ei have a mean of zero Engineering Statistics 22

  23. Correlation Analysis And Linear Regression Chapter 7 Least Squares Criterion • b0 and b1 are obtained by finding the values of b0 and b1 that minimize the sum of the squared residuals Engineering Statistics 23

  24. Correlation Analysis And Linear Regression Chapter 7 The Least Squares Equation • The formulas for b1 and b0 are: algebraic equivalent: and Engineering Statistics 24

  25. Correlation Analysis And Linear Regression Chapter 7 Interpretation of the Slope and the Intercept • b0 is the estimated average value of y when the value of x is zero • b1 is the estimated change in the average value of y as a result of a one-unit change in x Engineering Statistics 25

  26. Correlation Analysis And Linear Regression Chapter 7 Example 1 Data from a sample of 10 pharmacies are used to examine the relation between prescription sales volume and the percentage of prescription ingredients purchased from supplier. The sample data are shown below Engineering Statistics 26

  27. Correlation Analysis And Linear Regression Chapter 7 Solution:- May 19 Engineering Statistics 27

  28. Correlation Analysis And Linear Regression Chapter 7 Solution Engineering Statistics 28

  29. Correlation Analysis And Linear Regression Chapter 7 Example 2 • Suppose an appliance store conducts a 5-month experiment to determine the effect of advertising on sales revenue and obtains the following results Find the sample regression line and predict the sales revenue if the appliance store spends 4.5 thousand dollars for advertising in a month. Engineering Statistics 29

  30. Correlation Analysis And Linear Regression Chapter 7 Solution:- May 19 Engineering Statistics 30

  31. Correlation Analysis And Linear Regression Chapter 7 Solution:- May 19 Engineering Statistics 31

  32. Correlation Analysis And Linear Regression Chapter 7 Example 3 • Assume we have the following observations • Is there a relationship between x and y? • We propose a model on the form ^y = a + bx • How do we estimate a and b? May 19 Engineering Statistics 32

  33. Correlation Analysis And Linear Regression Chapter 7 Solution:- May 19 Engineering Statistics 33

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