Lesson 4

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Introductory Business Statistics
International Business School
Hanze University of Applied Sciences

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Lesson 4

  1. 1. Hanze University of Applied Science GroningenNing Ding, PhDLecturer of International BusinessSchool (IBS)n.ding@pl.hanze.nl
  2. 2. What we are going to learn?• Review• Chapter 12: Simple Regression and Correlation – dependent / independent variables – scatter diagrams – regression analysis – Least-squares estimating equation – the coefficient of determination – the coefficient of correlation
  3. 3. Review• Review What is the interquartile range? a. 98 b. 1764 c. 854 d.484 e.1940 f.2038• Chapter 12:Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams–regression analysis–Least-squares Interquartile Rangeestimating equation 3-Q1 =Q–the coefficient of =2205-1721determination =484–the coefficient ofcorrelation
  4. 4. Review• Review• Chapter 12: L=(8+1)*25%=2.25Simple Regressionand Correlation RangeInterquartile Q1=133.5–dependent / =274.5-133.5independent =141 L=(8+1)*75%=6.75variables–scatter diagrams–regression analysis Q3=274.5–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  5. 5. Median Quartile Decile Percentile 1 1 1st D 2 2 Q1=2 2 2 4 4 Interquartile 5 5 Range 7 7 8 8 Q3=8.5 9 9 12 12 9th DBoxplot How to interpret? http://cnx.org/content/m11192/latest/
  6. 6. Review a. Positive b. Negative c. Symmetrical d. No idea a b Mean= € 450€ 20 € 2000 Q1= € 250 Median= € 350 Q3= € 850 The distribution is skewed to __________ because the mean is the right larger than __________the median. http://cnx.org/content/m11192/latest/
  7. 7. 0.81.0 Mean > Median1.01.21.21.31.51.72.02.02.12.2 2.04.0 Mean < Median 3.2 Positively skewed 3.6 3.7 4.0 4.2 4.2 4.5 4.5 4.6 4.8http://qudata.com/online/statcalc/ 5.0 Negatively skewed 5.0
  8. 8. • Review This means that the data is• Chapter 12: symmetrically distributed.Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation Zero skewness mode=median=mean
  9. 9. Regression Analysis• Review• Chapter 12: – scatter diagramsSimple Regressionand Correlation – dependent / independent variables–dependent /independent – regression analysisvariables–scatter diagrams – Least-squares estimating equation–regression analysis–Least-squares – the coefficient of determinationestimating equation–the coefficient of – the coefficient of correlationdetermination–the coefficient ofcorrelation
  10. 10. Scatter Diagram• Review –How to determine both the nature and the• Chapter 12: strength of a relationship between variables.Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  11. 11. Describing Relationshi Scatter Diagram• Review Variables – Scatter Diag Scatter Diagram:• Chapter 12: StreudiagrammSimple Regression Puntenwolkand Correlation 散布图–dependent /independentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation Positive correlation
  12. 12. Scatter Diagram• Review• Chapter 12:Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation Negative correlation
  13. 13. catter DiagramDiagram Scatter Examples• Review• Chapter 12:Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation No correlation
  14. 14. Scatter Diagram• Review Scatter Diagrams:• Chapter 12: • Patterns indicating that the variables are relatedSimple Regressionand Correlation • If related, we can describe the relationship–dependent /independentvariables–scatter diagrams–regression analysisStrong & Positive Weak & Positive–Least-squares correlation correlationestimating equation–the coefficient ofdetermination–the coefficient of Nocorrelation correlation Weak & Negative Strong & Negative correlation correlation
  15. 15. Dependent/Independent Variables Describing Relations Variables: Variables – known• Review – Independent variables: Scatter D• Chapter 12:Simple Regression – Dependent variables: to predictand Correlation–dependent /independent Dependent Variablevariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation Independent Variable
  16. 16. Regression Analysis Correlation & Cause Effect?• Review • The relationships found by regression to be• Chapter 12: relationships of associationSimple Regressionand Correlation • Not necessarilly of cause and effect.–dependent /independentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  17. 17. Regression Analysis• Review• Chapter 12:Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  18. 18. Least-squares Estimating Equation• Review Least-squares estimating equation:• Chapter 12: • The dependent variable Y is determined by the independentSimple Regression variable Xand Correlation–dependent / Dependent Variableindependentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient of Ycorrelation X I 88 ? Independent Variable Ŷ = a + bX
  19. 19. Least-squares Estimating Equation• Review Least-squares estimating equation:• Chapter 12:Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams Ŷ = a + bX–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  20. 20. Least-squares Estimating Equation• Review Least-squares estimating equation:• Chapter 12:Simple Regressionand Correlation xy - n x y–dependent /independentvariables b= 2 2–scatter diagrams–regression analysis x -nx–Least-squaresestimating equation–the coefficient ofdetermination Y = a + bX a = Y - bX–the coefficient ofcorrelation
  21. 21. Least-squares Estimating Equation the relationship between the age of a truck and the annual repair expense?• Review xy - nx y b= Y = a + bX a = Y - bX x -nx 2 2 Step 2:• Chapter 12:Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation Step 1: X=3 Y=6–the coefficient ofdetermination 78 - 4 * 3 * 6–the coefficient of Step 4: b= 0.75 Step 6: Ŷ = 3.75 + 0.75 Xcorrelation 44 - 4 * 9 a = 6 - 0.75*3 = 3.75 Step 7: 6.75 = 3.75 + 0.75 * 4 Step 5: If the city has a truck that is 4 years old, Step 8: the director could use the equation to predict $675 annually in repairs.
  22. 22. Least-squares Estimating Equation• Review To find the simple/linear regression of Personal Income (X) and Auto Sales (Y)• Chapter 12:Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams–regression analysis If X=64, what about Y?–Least-squaresestimating equation–the coefficient of Step 1: Count the number of values. N = 5determination a. 4.1 Step 2: Find XY, X2 See the below table–the coefficient of b. 5.3correlation c. 6.7 d. 7.4 e. 7.5 f. 8.2
  23. 23. Least-squares Estimating Equation• Review• Chapter 12:Simple Regressionand Correlation–dependent /independentvariables–scatter diagrams Step 3: Find ΣX, ΣY, ΣXY, ΣX2.–regression analysis ΣX = 311 Mean = 62.2–Least-squares ΣY = 18.6 Mean = 3.72estimating equation ΣXY = 1159.7–the coefficient of ΣX2 = 19359determination–the coefficient ofcorrelation Step 4: xy - nx y b= -nx 2 2 x Substitute in the above slope formula given. Slope(b) = 1159.7-5*62.2*3.72 = 0.19 19359-5*62.2*62.2
  24. 24. Least-squares Estimating Equation• Review Slope(b) = 0.19• Chapter 12: Now, again substitute in the above intercept formula given. Step 5:Simple Regressionand Correlation Intercept(a) = Y - bX = 3.72- 0.19 * 62.2= -8.098–dependent /independent Step 6:variables Then substitute these values in regression equation–scatter diagrams formula–regression analysis Regression Equation(Ŷ) = a + bX–Least-squaresestimating equation Ŷ = -8.098 + 0.19X–the coefficient ofdetermination Regression Equation:–the coefficient of Suppose if we want to know the Ŷ = a + bXcorrelation approximate y value for the variable X = -8.098 + 0.19(64) = -8.098 + 12.16 = 64. Then we can substitute the value = 4.06 in the above equation.
  25. 25. Standard Error• Review Standard Error: to minimize the sum of the squares of the errors to measure the• Chapter 12: goodness of fit of a lineSimple Regressionand Correlation–dependent / SE SEindependent ei = residualivariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation Strong Weak correlation correlation
  26. 26. Standard Error• Review• Chapter 12: ei = residualiSimple Regressionand Correlation–dependent /independentvariables–scatter diagrams–regression analysis–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  27. 27. Coefficient of Determination• Review Correlation Analysis:• Chapter 12: describe the degree to which one variable is linearlySimple Regression related to another.and Correlation–dependent /independentvariables Coefficient of Determination: r 2–scatter diagrams–regression analysis Measure the extent, or strength, of the association that–Least-squaresestimating equation exists between two variables.–the coefficient ofdetermination–the coefficient ofcorrelation Coefficient of Correlation: r Square root of coefficient of determination
  28. 28. Coefficient of Determination Coefficient of Determination:• Review r2• Chapter 12: • 0 ≤ r2 ≤ 1.Simple Regression • The larger r2 , the stronger the linear relationship.and Correlation–dependent / • The closer r2 is to 1, the more confident we are inindependent our prediction.variables–scatter diagrams–regression analysis–Least-squaresestimating equation r 2=0.9984–the coefficient ofdetermination–the coefficient ofcorrelation
  29. 29. Coefficient of Determination• Review• Chapter 12: • 76.30% of Sales changes is explained bySimple Regression GDP changes. The rest 23.70% isand Correlation–dependent / explained by other variables.independentvariables–scatter diagrams–regression analysis–Least-squares r 2=0.7630estimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  30. 30. Coefficient of Correlation• Coefficient of correlation: Review r• Chapter 12: • r ≤ 0.3 Weak CorrelationSimple Regressionand Correlation • 0.3 ≤ r ≤ 0.7 Moderate Correlation–dependent / • r ≥ 0.7 Strong Correlationindependentvariables • r = 0.10 Perfect Correlation r 2=0.1132–scatter diagrams–regression analysis–Least-squares r =0.1064estimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  31. 31. Coefficient of Correlation• Review • There is a positive and weak correlation• Chapter 12: r between GDP and Envy Rides’ annualSimple Regression sales.and Correlation • 11.32% of Sales changes is explained by–dependent /independent r 2 GDP changes. The rest 88.68% isvariables r 2=0.1132–scatter diagrams explained by other variables.–regression analysis–Least-squares r =0.1064estimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  32. 32. Coefficient of Correlation• Review • There is a positive and strong correlation r between GDP and Envy Rides’ annual• Chapter 12:Simple Regression sales.and Correlation • 76.30% of Sales changes is explained by–dependent / r 2 GDP changes. The rest 23.70% isindependentvariables explained by other variables. r 2=0.7630–scatter diagrams–regression analysis–Least-squares r =0.8735estimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  33. 33. Coefficient of Correlation• Review • There is a positive and almost perfect• Chapter 12: r correlation between GDP and Envy Rides’Simple Regression annual sales.and Correlation • 99.84% of Sales changes is explained by–dependent /independent r 2 GDP changes. The rest 8% is explained byvariables other variables.–scatter diagrams r 2=0.9984–regression analysis–Least-squaresestimating equation r =0.9992–the coefficient ofdetermination–the coefficient ofcorrelation
  34. 34. Review• Review• Chapter 12:Simple Regression Which value of r indicates a stronger correlation than 0.40?and Correlation–dependent / A. -0.30independent B. -0.50variables C. +0.38–scatter diagrams D. 0–regression analysis–Least-squaresestimating equation If all the plots on a scatter diagram lie on a straight line, what is the–the coefficient of standard error of estimate?determination A. -1–the coefficient ofcorrelation B. +1 C. 0 D. Infinity
  35. 35. Review• Review• Chapter 12:Simple Regression In the least squares equation, Ŷ = 10 + 20X the value of 20and Correlation indicates–dependent / A. the Y intercept.independentvariables B. for each unit increase in X, Y increases by 20.–scatter diagrams C. for each unit increase in Y, X increases by 20.–regression analysis D. none of these.–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  36. 36. Review• Review A sales manager for an advertising agency believes there is a• Chapter 12: relationship between the number of contacts and the amount of theSimple Regression sales. To verify this belief, the following data was collected:and Correlation–dependent / What is the Y-intercept of the linear equation?independent A. -12.201variables B. 2.1946–scatter diagrams C. -2.1946–regression analysis D. 12.201–Least-squaresestimating equation–the coefficient ofdetermination–the coefficient ofcorrelation
  37. 37. What we have learnt?• Review• Chapter 12: – scatter diagramsSimple Regressionand Correlation – dependent / independent variables–dependent /independent – regression analysisvariables–scatter diagrams – Least-squares estimating equation–regression analysis–Least-squares – the coefficient of determinationestimating equation–the coefficient of – the coefficient of correlationdetermination–the coefficient ofcorrelation

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