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### 004

1. 1. IBS Statistics Year 1 Dr. Ning DING
2. 2. Table of content <ul><li>Review </li></ul><ul><ul><li>Interquartile Range </li></ul></ul><ul><ul><li>Skewness </li></ul></ul><ul><li>Learning Goals </li></ul><ul><li>Chapter 12: Simple Regression and Correlation </li></ul><ul><li>Exercises </li></ul>
3. 3. Review Chapter 3: Describing Data Find the interquartile range:   1460 1471 1637 1721 1758 1787 1940 2038 2047 2054 2097 2205 2287 2311 2406 Interquartile Range =Q 3 -Q 1 =2205-1721 =484
4. 4. Correction of EXCEL Exercise 5 L=(8+1)*25%=2.25 Q1=133.5 L=(8+1)*75%=6.75 Q3=274.5 Interquartile Range =274.5-133.5 =141
5. 5. Boxplot 1 2 2 4 5 7 8 9 12 Median 1 2 2 4 7 8 9 12 Quartile Q 1 =2 Q 3 =8.5 5 Interquartile Range Decile 1st D 9th D Percentile http://cnx.org/content/m11192/latest/ How to interpret?
6. 6. Boxplot The distribution is skewed to __________ because the mean is __________the median. the right larger than http://cnx.org/content/m11192/latest/ € 20 € 2000 Q 1 = € 250 Q 3 = € 850 Median= € 350 Mean= € 450 a b
7. 7. 0.8 1.0 1.0 1.2 1.2 1.3 1.5 1.7 2.0 2.0 2.1 2.2 4.0 2.0 3.2 3.6 3.7 4.0 4.2 4.2 4.5 4.5 4.6 4.8 5.0 5.0 Mean > Median Mean < Median Positively skewed Negatively skewed http://qudata.com/online/statcalc/
8. 8. This means that the data is symmetrically distributed . Zero skewness mode=median=mean
9. 9. Learning Goals <ul><li>Chapter 12: </li></ul><ul><ul><li>Learn how many business decisions depend on knowing the specific relationship between two or more variables </li></ul></ul><ul><ul><li>Use scatter diagrams to visualize the relationship between two variables </li></ul></ul><ul><ul><li>Use regression analysis to estimate the relationship between two variables </li></ul></ul><ul><ul><li>Use the least-squares estimating equation to predict future values of the dependent variable </li></ul></ul><ul><ul><li>Learn how correlation analysis describes the degree to which two variables are linearly related to each other </li></ul></ul><ul><ul><li>Understand the coefficient of determination as a measure of the strength of the relationship between two variables </li></ul></ul><ul><ul><li>Learn limitations of regression and correlation analyses and caveats about their use. </li></ul></ul>
10. 10. 1. Introduction Chapter 12: Sim Reg & Corr Regression and Correlation Analyses: <ul><ul><li>How to determine both the nature and the strength of a relationship between variables. </li></ul></ul>
11. 11. 1. Introduction Chapter 12: Sim Reg & Corr Scatter Diagram: Positive correlation
12. 12. 1. Introduction Chapter 12: Sim Reg & Corr Scatter Diagram: Negative correlation
13. 13. 1. Introduction Chapter 12: Sim Reg & Corr Scatter Diagram: No correlation
14. 14. 2. Types of Relationships Chapter 12: Sim Reg & Corr Variables: <ul><ul><li>Independent variables: known </li></ul></ul><ul><ul><li>Dependent variables: to predict </li></ul></ul>Independent Variable Dependent Variable
15. 15. 2. Types of Relationship Chapter 12: Sim Reg & Corr <ul><li>Correlation & Cause Effect? </li></ul><ul><li>The relationships found by regression to be relationships of association </li></ul><ul><li>Not necessarilly of cause and effect. </li></ul>Independent Variable Dependent Variable
16. 16. 2. Estimation Using the Regression Line Chapter 12: Sim Reg & Corr <ul><li>Scatter Diagrams: </li></ul><ul><li>Patterns indicating that the variables are related </li></ul><ul><li>If related, we can describe the relationship </li></ul>Strong & Positive correlation Strong & Negative correlation Weak & Positive correlation Weak & Negative correlation No correlation
17. 17. Chapter 12: Sim Reg & Corr Scatter Diagrams: 2. Estimation Using the Regression Line
18. 18. 2. Estimation Using the Regression Line Chapter 12: Sim Reg & Corr <ul><li>Simple Linear Regression: </li></ul><ul><li>The dependent variable Y is determined by the independent variable X </li></ul>Ŷ = a + b X Independent Variable Dependent Variable Y X
19. 19. 2. Estimation Using the Regression Line Chapter 12: Sim Reg & Corr <ul><li>Simple Linear Regression: </li></ul><ul><li>The dependent variable Y is determined by the independent variable X </li></ul>Ŷ = a + b X
20. 20. 2. Estimation Using the Regression Line Chapter 12: Sim Reg & Corr Slope of the Best-Fitting Regression Line: Y = a + b X a = Y - b X
21. 21. 2. Estimation Using the Regression Line Chapter 12: Sim Reg & Corr the relationship between the age of a truck and the annual repair expense? a = 6 - 0.75*3 = 3.75 Ŷ = 3.75 + 0.75 X If the city has a truck that is 4 years old, the director could use the equation to predict \$675 annually in repairs. 6.75 = 3.75 + 0.75 * 4 Y = a + b X a = Y - b X X=3 Y=6
22. 22. Exercise Chapter 12: Sim Reg & Corr <ul><li>Example: </li></ul><ul><li>To find the simple/linear regression of Personal Income ( X ) and Auto Sales ( Y ) </li></ul>Count the number of values.       Find XY, X 2   See the below table N = 5 X=64 what about Y? Step 1: Step 2:
23. 23. Exercise Chapter 12: Sim Reg & Corr Find Σ X, Σ Y, Σ XY, Σ X 2 .             Σ X = 311 Mean = 62.2             Σ Y = 18.6 Mean = 3.72             Σ XY = 1159.7             Σ X 2 = 19359 Step 3: Step 4: Substitute in the above slope formula given.             Slope(b) = = 0.19 1159.7-5*62.2*3.72 19359-5*62.2*62.2
24. 24. Exercise Chapter 12: Sim Reg & Corr Then substitute these values in regression equation formula             Regression Equation( Ŷ ) = a + bX           Ŷ   = -8.098 + 0.19 X .             Slope(b) = 0.19 Suppose if we want to know the approximate y value for the variable X = 64. Then we can substitute the value in the above equation. Regression Equation: Ŷ = a + bX             = -8.098 + 0.19( 64 ).             = -8.098 + 12.16             = 4.06 Step 5: Step 6: Now, again substitute in the above intercept formula given.             Intercept(a) = Y - b X   = 3.72- 0.19 * 62.2= -8.098
25. 25. 2. Estimation Using the Regression Line Chapter 12: Sim Reg & Corr Least Squares Method: Minimize the sum of the squares of the errors to measure the goodness of fit of a line e i = residual i
26. 26. 2. Estimation Using the Regression Line Chapter 12: Sim Reg & Corr Least Squares Method:
27. 27. 2. Estimation Using the Regression Line Chapter 12: Sim Reg & Corr Example:
28. 28. 2. Estimation Using the Regression Line Chapter 12: Sim Reg & Corr Example Solution:
29. 29. 3. Correlation Analysis Chapter 12: Sim Reg & Corr Correlation Analysis: describe the degree to which one variable is linearly related to another. Coefficient of Determination: Measure the extent, or strength, of the association that exists between two variables. Coefficient of Correlation: Square root of coefficient of determination r 2 r
30. 30. 3. Correlation Analysis Chapter 12: Sim Reg & Corr Coefficient of Determination: Measure the extent, or strength, of the association that exists between two variables. <ul><li>0 ≤ r 2 ≤ 1. </li></ul><ul><li>The larger r 2 , the stronger the linear relationship. </li></ul><ul><li>The closer r 2 is to 1, the more confident we are in our prediction. </li></ul>
31. 31. 3. Correlation Analysis Chapter 12: Sim Reg & Corr Coefficient of Correlation:
32. 32. 3. Correlation Analysis Chapter 12: Sim Reg & Corr Coefficient of Determination:
33. 33. 3. Correlation Analysis Chapter 12: Sim Reg & Corr Example Solution:
34. 34. 3. Correlation Analysis Chapter 12: Sim Reg & Corr Example Solution:
35. 35. Review Chapter 3: Describing Data Which value of r indicates a stronger correlation than 0.40?  A. -0.30 B. -0.50 C. +0.38 D. 0 If all the plots on a scatter diagram lie on a straight line, what is the standard error of estimate?  A. -1 B. +1 C. 0 D. Infinity
36. 36. Review Chapter 3: Describing Data In the least squares equation,   Ŷ  = 10 + 20 X the value of 20 indicates  A. the Y intercept. B. for each unit increase in X , Y increases by 20. C. for each unit increase in Y , X increases by 20. D. none of these.
37. 37. Exercise Chapter 3: Describing Data A sales manager for an advertising agency believes there is a relationship between the number of contacts and the amount of the sales. To verify this belief, the following data was collected:   What is the Y-intercept of the linear equation?  A. -12.201 B. 2.1946 C. -2.1946 D. 12.201
38. 38. Exercise Chapter 12: Sim Reg & Corr Ŷ = -1.8182 + 0.1329X Sample Exam P.4
39. 39. Exercise Chapter 12: Sim Reg & Corr Sample Exam P.4
40. 40. Exercise Chapter 12: Sim Reg & Corr Sample Exam P.4 Ŷ = -1.8182 + 0.1329X
41. 41. Summary Chapter 1: What is Statistics? <ul><li>Chapter 3: </li></ul><ul><ul><li>Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean </li></ul></ul><ul><ul><li>Explain the characteristics, uses, advantages, and disadvantages of each measure of location </li></ul></ul><ul><ul><li>Identify the position of the mean, median, and mode for both symmetric and skewed distributions </li></ul></ul><ul><ul><li>Compute and interpret the range, mean deviation, variance, and standard deviation </li></ul></ul><ul><ul><li>Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion </li></ul></ul><ul><ul><li>Understand Chebyshev’s theorem and the Empirical Rule as they relate to a set of observations </li></ul></ul>