Basic Business Statistics 6 th Edition

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1 Basic Business Statistics 6 th Edition Chapter 12 Simple Linear Regression

2 Learning Objectives In this chapter, you learn: How to use regression analysis to predict the value of a dependent variable based on an independent variable The meaning of the regression coefficients b 0 and b 1 How to evaluate the assumptions of regression analysis and know what to do if the assumptions are violated To make inferences about the slope and correlation coefficient To estimate mean values and predict individual values Chap 12-2

3 Correlation vs. Regression A scatter plot can be used to show the relationship between two variables Correlation analysis is used to measure the strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the relationship No causal effect is implied with correlation Scatter plots were first presented in Ch. 2 Correlation was first presented in Ch. 3 Chap 12-3

4 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 predict or explain Independent variable: the variable used to predict or explain the dependent variable Chap 12-4

5 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 related to changes in X Chap 12-5

6 Types of Relationships Linear relationships Curvilinear relationships Y Y X X Y Y X X Chap 12-6

7 Types of Relationships (continued) Strong relationships Weak relationships Y Y X X Y Y X X Chap 12-7

8 Types of Relationships No relationship (continued) Y X Y X Chap 12-8

9 Simple Linear Regression Model Dependent Variable Y Population Y intercept i β 0 Population Slope Coefficient β 1 Independent Variable X i ε i Random Error term Linear component Random Error component Chap 12-9

10 Simple Linear Regression Model (continued) Y Y i β 0 β 1 X i ε i Observed Value of Y for X i Predicted Value of Y for X i Intercept = β 0 ε i Random Error for this X i value Slope = β 1 X i X Chap 12-10

11 Simple Linear Regression Equation (Prediction Line) The simple linear regression equation provides an estimate of the population regression line Estimated (or predicted) Y value for observation i Estimate of the regression intercept Estimate of the regression slope Ŷ i b 0 b 1 X i Value of X for observation i Chap 12-11

12 The Least Squares Method b 0 and b 1 are obtained by finding the values of X and Y that minimize the sum of the squared differences between Y and Ŷ. min (Y Ŷ ) 2 min (Y (b b X )) 2 i i i 0 1 i Chap 12-12

13 Finding the Least Squares Equation The coefficients b 0 and b 1, and other regression results in this chapter, can be found using Excel or Minitab Formulas are shown on the key equations sheet and in the text Chap 12-13

14 Interpretation of the Slope and the Intercept b 0 is the estimated mean value of Y when the value of X is zero b 1 is the estimated change in the mean value of Y as a result of a one-unit increase in X Chap 12-14

15 Simple Linear Regression Example A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet) A random sample of 10 houses is selected Dependent variable (Y) = house price in $1000s Independent variable (X) = square feet Chap 12-15

16 Simple Linear Regression Example: Data House Price in $1000s (Y) Square Feet (X) Chap 12-16

17 Simple Linear Regression Example: Scatter Plot House price model: Scatter Plot House Price ($1000s) Square Feet Chap 12-17

18 Simple Linear Regression Example: Using Excel Data Analysis Function 1. Choose Data 2. Choose Data Analysis 3. Choose Regression Chap 12-18

19 Simple Linear Regression Example: Using Excel Data Analysis Function (continued) Enter Y s and X s and desired options Chap 12-19

20 Simple Linear Regression Example: Using PHStat2 Add-Ins: PHStat2: Regression: Simple Linear Regression

21 Simple Linear Regression Example: Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 10 The regression equation is: house price (square feet) ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet Chap 12-21

22 Simple Linear Regression Example: Graphical Representation House price model: Scatter Plot and Prediction Line Intercept = House Price ($1000s) Slope = Square Feet house price (square feet) Chap 12-22

23 Simple Linear Regression Example: Interpretation of b o house price (square feet) b 0 is the estimated mean value of Y when the value of X is zero (if X = 0 is in the range of observed X values) Because a house cannot have a square footage of 0, b 0 has no practical application Chap 12-23

24 Simple Linear Regression Example: Interpreting b 1 house price (square feet) b 1 estimates the change in the mean value of Y as a result of a one-unit increase in X Here, b 1 = tells us that the mean value of a house increases by.10977($1000) = $109.77, on average, for each additional one square foot of size Chap 12-24

25 Simple Linear Regression Example: Making Predictions Predict the price for a house with 2000 square feet: house price (sq.ft.) (2000) The predicted price for a house with 2000 square feet is ($1,000s) = $317,780 Chap 12-25

26 Simple Linear Regression Example: Making Predictions When using a regression model for prediction, only predict within the relevant range of data Relevant range for interpolation House Price ($1000s) Square Feet Do not try to extrapolate beyond the range of observed X s Chap 12-26

27 Measures of Variation Total variation is made up of two parts: SST SSR SSE Total Sum of Squares Regression Sum of Squares Error Sum of Squares 2 SST (Y Y 2 SSR (Ŷ Y SSE (Y i ) i ) 2 i Ŷi ) where: Y = Mean value of the dependent variable Y i = Observed value of the dependent variable Yˆi = Predicted value of Y for the given X i value Chap 12-27

28 Measures of Variation SST = total sum of squares (Total Variation) Measures the variation of the Y i values around their mean Y SSR = regression sum of squares (Explained Variation) Variation attributable to the relationship between X and Y SSE = error sum of squares (Unexplained Variation) Variation in Y attributable to factors other than X (continued) Chap 12-28

29 Measures of Variation (continued) Y Y i _ Y Y _ SST = (Y i - Y) 2 SSE = (Y i - Y i ) 2 _ SSR = (Y i - Y) 2 Y _ Y X i X Chap 12-29

30 Coefficient of Determination, r 2 The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable The coefficient of determination is also called r-squared and is denoted as r 2 r 2 SSR SST regression sum of squares total sum of squares note: 0 r 2 1 Chap 12-30

31 Y Examples of Approximate r 2 Values r 2 = 1 Y r 2 = 1 X Perfect linear relationship between X and Y: 100% of the variation in Y is explained by variation in X r 2 = 1 X Chap 12-31

32 Examples of Approximate r 2 Values Y 0 < r 2 < 1 Y X Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X X Chap 12-32

33 Examples of Approximate r 2 Values Y r 2 = 0 No linear relationship between X and Y: r 2 = 0 X The value of Y does not depend on X. (None of the variation in Y is explained by variation in X) Chap 12-33

34 Simple Linear Regression Example: Coefficient of Determination, r 2 in Excel Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 10 ANOVA df SS MS F Significance F Regression Residual Total r 2 SSR SST % of the variation in house prices is explained by variation in square feet Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet Chap 12-34

35 Standard Error of Estimate The standard deviation of the variation of observations around the regression line is estimated by S Where YX SSE n 2 i1 SSE = error sum of squares n = sample size n ( Y n i Yˆ 2 i ) 2 Chap 12-35

36 Simple Linear Regression Example: Standard Error of Estimate in Excel Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 10 S YX ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet Chap 12-36

37 Comparing Standard Errors S YX is a measure of the variation of observed Y values from the regression line Y Y smalls YX X large S YX X The magnitude of S YX should always be judged relative to the size of the Y values in the sample data i.e., S YX = $41.33K is moderately small relative to house prices in the $200K - $400K range Chap 12-37

38 Assumptions of Regression L.I.N.E Linearity The relationship between X and Y is linear Independence of Errors Error values are statistically independent Normality of Error Error values are normally distributed for any given value of X Equal Variance (also called homoscedasticity) The probability distribution of the errors has constant variance Chap 13-38

39 Residual Analysis e The residual for observation i, e i, is the difference between its observed and predicted value Check the assumptions of regression by examining the residuals Examine for linearity assumption Evaluate independence assumption Evaluate normal distribution assumption Examine for constant variance for all levels of X (homoscedasticity) Graphical Analysis of Residuals i Y i Ŷ i Can plot residuals vs. X Chap 12-39

40 Residual Analysis for Linearity Y Y x x residuals x residuals x Not Linear Linear Chap 12-40

41 Residual Analysis for Independence Not Independent Independent residuals X residuals X residuals X Chap 12-41

42 Checking for Normality Examine the Stem-and-Leaf Display of the Residuals Examine the Boxplot of the Residuals Examine the Histogram of the Residuals Construct a Normal Probability Plot of the Residuals Chap 12-42

43 Residual Analysis for Normality When using a normal probability plot, normal errors will approximately display in a straight line Percent Residual Chap 12-43

44 Residual Analysis for Equal Variance Y Y x x residuals x residuals x Non-constant variance Constant variance Chap 12-44

45 Simple Linear Regression Example: Excel Residual Output RESIDUAL OUTPUT Predicted House Price Residuals 80 House Price Model Residual Plot Residuals Square Feet Does not appear to violate any regression assumptions Chap 12-45

46 Simple Linear Regression Example: Minitab Residual Output Percent Normal Probability Plot Residual Plots for House Price (Y) Residual Versus Fits Residual Fitted Value Histogram Versus Order 3 50 Frequency 2 1 Residual Residual Observation Order Chap 13-46

47 Inferences About the Slope The standard error of the regression slope coefficient (b 1 ) is estimated by Sb 1 SYX SSX S (X YX i X) 2 where: S b1 = Estimate of the standard error of the slope S YX SSE n 2 = Standard error of the estimate Chap 12-47

48 Inferences About the Slope: t Test t test for a population slope Is there a linear relationship between X and Y? Null and alternative hypotheses H 0 : β 1 = 0 (no linear relationship) H 1 : β 1 0 (linear relationship does exist) Test statistic t STAT d.f. n b 1 S 2 b 1 β 1 where: b 1 = regression slope coefficient β 1 = hypothesized slope S b1 = standard error of the slope Chap 12-48

49 Inferences About the Slope: t Test Example House Price in $1000s (y) Square Feet (x) Estimated Regression Equation: house price (sq.ft.) The slope of this model is Is there a relationship between the square footage of the house and its sales price? Chap 12-49

50 Inferences About the Slope: t Test Example From Excel output: H 0 : β 1 = 0 H 1 : β 1 0 Coefficients Standard Error t Stat P-value Intercept Square Feet From Minitab output: b 1 S b1 Predictor Coef SE Coef T P Constant Square Feet b 1 S b 1 t STAT b 1 S β b Chap 12-50

51 Inferences About the Slope: t Test Example Test Statistic: t STAT = H 0 : β 1 = 0 H 1 : β 1 0 d.f. = 10-2 = 8 /2=.025 /2=.025 Decision: Reject H 0 Reject H 0 Reject H 0 Do not reject H -t 0 α/2 t 0 α/ There is sufficient evidence that square footage affects house price Chap 12-51

52 Inferences About the Slope: t Test Example From Excel output: H 0 : β 1 = 0 H 1 : β 1 0 Coefficients Standard Error t Stat P-value Intercept Square Feet From Minitab output: Predictor Coef SE Coef T P Constant Square Feet Decision: Reject H 0, since p-value < α There is sufficient evidence that square footage affects house price. p-value Chap 12-52

53 Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: b1 t α / 2S b 1 d.f. = n - 2 Excel Printout for House Prices: Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet At 95% level of confidence, the confidence interval for the slope is (0.0337, ) Chap 12-53

54 Confidence Interval Estimate for the Slope (continued) Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.74 and $ per square foot of house size This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the.05 level of significance Chap 12-54

55 Estimating Mean Values and Predicting Individual Values Goal: Form intervals around Y to express uncertainty about the value of Y for a given X i Confidence Interval for the mean of Y, given X i Y = b 0 +b 1 X i Y Y Prediction Interval for an individual Y, given X i X i X Chap 12-55

56 Confidence Interval for the Average Y, Given X Confidence interval estimate for the mean value of Y given a particular X i Confidence interval for μ Yˆ t α / 2 S YX h i Y XX i : Size of interval varies according to distance away from mean, X h i (Xi X) 1 (Xi X) 2 n SSX n (X X) i Chap 12-56

57 Prediction Interval for an Individual Y, Given X Confidence interval estimate for an Individual value of Y given a particular X i Confidence interval for Y Yˆ t α / 2 S YX 1 XX h i i : This extra term adds to the interval width to reflect the added uncertainty for an individual case Chap 12-57

58 Estimation of Mean Values: Example Confidence Interval Estimate for μ Y X=X Find the 95% confidence interval for the mean price of 2,000 square-foot houses Predicted Price Y i = ($1,000s) i 2 1 (Xi X) t 0.025SYX n (X X) Ŷ 2 i The confidence interval endpoints are and , or from $280,660 to $354,900 Chap 12-58

59 Estimation of Individual Values: Example Prediction Interval Estimate for Y X=X Find the 95% prediction interval for an individual house with 2,000 square feet Predicted Price Y i = ($1,000s) 2 1 (Xi X) t 0.025SYX n (X X) Ŷ 2 i The prediction interval endpoints are and , or from $215,500 to $420,070 i Chap 12-59

60 Finding Confidence and Prediction Intervals in Excel From Excel, use PHStat regression simple linear regression Check the confidence and prediction interval for X= box and enter the X-value and confidence level desired Chap 12-60

61 Finding Confidence and Prediction Intervals in Excel (continued) Input values Y Confidence Interval Estimate for μ Y X=Xi Prediction Interval Estimate for Y X=Xi Chap 12-61

62 Pitfalls of Regression Analysis Lacking an awareness of the assumptions underlying least-squares regression Not knowing how to evaluate the assumptions Not knowing the alternatives to least-squares regression if a particular assumption is violated Using a regression model without knowledge of the subject matter Extrapolating outside the relevant range Chap 12-62

63 Strategies for Avoiding the Pitfalls of Regression Start with a scatter plot of X vs. Y to observe possible relationship Perform residual analysis to check the assumptions Plot the residuals vs. X to check for violations of assumptions such as homoscedasticity Use a histogram, stem-and-leaf display, boxplot, or normal probability plot of the residuals to uncover possible non-normality Chap 12-63

64 Strategies for Avoiding the Pitfalls of Regression (continued) If there is violation of any assumption, use alternative methods or models If there is no evidence of assumption violation, then test for the significance of the regression coefficients and construct confidence intervals and prediction intervals Avoid making predictions or forecasts outside the relevant range Chap 12-64

65 Chapter Summary Introduced types of regression models Reviewed assumptions of regression and correlation Discussed determining the simple linear regression equation Described measures of variation Discussed residual analysis Chap 12-65

66 Chapter Summary (continued) Described inference about the slope Discussed correlation -- measuring the strength of the association Addressed estimation of mean values and prediction of individual values Discussed possible pitfalls in regression and recommended strategies to avoid them

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