1、Fall 2006 Fundamentals of Business Statistics,1,Chapter 13 Introduction to Linear Regression and Correlation Analysis,Fall 2006 Fundamentals of Business Statistics,2,Chapter Goals,To understand the methods for displaying and describing relationship among variables,Fall 2006 Fundamentals of Business
2、Statistics,3,Methods for Studying Relationships,Graphical Scatterplots Line plots 3-D plots Models Linear regression Correlations Frequency tables,Fall 2006 Fundamentals of Business Statistics,4,Two Quantitative Variables,The response variable, also called the dependent variable, is the variable we
3、want to predict, and is usually denoted by y. The explanatory variable, also called the independent variable, is the variable that attempts to explain the response, and is denoted by x.,Fall 2006 Fundamentals of Business Statistics,5,YDI 7.1,Fall 2006 Fundamentals of Business Statistics,6,Scatter Pl
4、ots and Correlation,A scatter 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 Only concerned with strength of the relationship No causal effect is implied,
5、Fall 2006 Fundamentals of Business Statistics,7,Example,The following graph shows the scatterplot of Exam 1 score (x) and Exam 2 score (y) for 354 students in a class. Is there a relationship?,Fall 2006 Fundamentals of Business Statistics,8,Scatter Plot Examples,y,x,y,x,y,y,x,x,Linear relationships,
6、Curvilinear relationships,Fall 2006 Fundamentals of Business Statistics,9,Scatter Plot Examples,y,x,y,x,No relationship,(continued),Fall 2006 Fundamentals of Business Statistics,10,Correlation Coefficient,The population correlation coefficient (rho) measures the strength of the association between t
7、he variablesThe sample correlation coefficient r is an estimate of and is used to measure the strength of the linear relationship in the sample observations,(continued),Fall 2006 Fundamentals of Business Statistics,11,Features of and r,Unit free Range between -1 and 1 The closer to -1, the stronger
8、the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker the linear relationship,Fall 2006 Fundamentals of Business Statistics,12,Examples of Approximate r Values,y,x,y,x,y,x,y,x,y,x,Tag with appropriate value: -1, -.6, 0, +.3, 1,Fal
9、l 2006 Fundamentals of Business Statistics,13,Earlier Example,Fall 2006 Fundamentals of Business Statistics,14,YDI 7.3,What kind of relationship would you expect in the following situations: age (in years) of a car, and its price.number of calories consumed per day and weight.height and IQ of a pers
10、on.,Fall 2006 Fundamentals of Business Statistics,15,YDI 7.4,Identify the two variables that vary and decide which should be the independent variable and which should be the dependent variable. Sketch a graph that you think best represents the relationship between the two variables. The size of a pe
11、rsons vocabulary over his or her lifetime. The distance from the ceiling to the tip of the minute hand of a clock hung on the wall.,Fall 2006 Fundamentals of Business Statistics,16,Introduction to Regression Analysis,Regression analysis is used to: Predict the value of a dependent variable based on
12、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,Fall 2006 Fundamentals of Business Statistic
13、s,17,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,Fall 2006 Fundamentals of Business Statistics,18,Types of Regression Models,Positive Linear Relationship,Negative
14、Linear Relationship,Relationship NOT Linear,No Relationship,Fall 2006 Fundamentals of Business Statistics,19,Linear component,Population Linear Regression,The population regression model:,Population y intercept,Population Slope Coefficient,Random Error term, or residual,Dependent Variable,Independen
15、t Variable,Random Errorcomponent,Fall 2006 Fundamentals of Business Statistics,20,Linear Regression Assumptions,Error values () are statistically independent Error values are normally distributed for any given value of x The probability distribution of the errors is normal The probability distributi
16、on of the errors has constant variance The underlying relationship between the x variable and the y variable is linear,Fall 2006 Fundamentals of Business Statistics,21,Population Linear Regression,(continued),Random Error for this x value,y,x,Observed Value of y for xi,Predicted Value of y for xi,xi
17、,Slope = 1,Intercept = 0,i,Fall 2006 Fundamentals of Business Statistics,22,The sample regression line provides an estimate of the population regression line,Estimated Regression Model,Estimate of the regression intercept,Estimate of the regression slope,Estimated (or predicted) y value,Independent
18、variable,The individual random error terms ei have a mean of zero,Fall 2006 Fundamentals of Business Statistics,23,Earlier Example,Fall 2006 Fundamentals of Business Statistics,24,Residual,A residual is the difference between the observed response y and the predicted response . Thus, for each pair o
19、f observations (xi, yi), the ith residual is ei = yi i = yi (b0 + b1x),Fall 2006 Fundamentals of Business Statistics,25,Least Squares Criterion,b0 and b1 are obtained by finding the values of b0 and b1 that minimize the sum of the squared residuals,Fall 2006 Fundamentals of Business Statistics,26,b0
20、 is the estimated average value of y when the value of x is zerob1 is the estimated change in the average value of y as a result of a one-unit change in x,Interpretation of the Slope and the Intercept,Fall 2006 Fundamentals of Business Statistics,27,The Least Squares Equation,The formulas for b1 and
21、 b0 are:,algebraic equivalent:,and,Fall 2006 Fundamentals of Business Statistics,28,Finding the Least Squares Equation,The coefficients b0 and b1 will usually be found using computer software, such as Excel, Minitab, or SPSS.Other regression measures will also be computed as part of computer-based r
22、egression analysis,Fall 2006 Fundamentals of Business Statistics,29,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) = hous
23、e price in $1000s Independent variable (x) = square feet,Fall 2006 Fundamentals of Business Statistics,30,Sample Data for House Price Model,Fall 2006 Fundamentals of Business Statistics,31,SPSS Output,The regression equation is:,Fall 2006 Fundamentals of Business Statistics,32,Graphical Presentation
24、,House price model: scatter plot and regression line,Slope = 0.110,Intercept = 98.248,Fall 2006 Fundamentals of Business Statistics,33,Interpretation of the Intercept, b0,b0 is the estimated average value of Y when the value of X is zero (if x = 0 is in the range of observed x values) Here, no house
25、s had 0 square feet, so b0 = 98.24833 just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet,Fall 2006 Fundamentals of Business Statistics,34,Interpretation of the Slope Coefficient, b1,b1 measures the estimated c
26、hange in the average value of Y as a result of a one-unit change in X Here, b1 = .10977 tells us that the average value of a house increases by .10977($1000) = $109.77, on average, for each additional one square foot of size,Fall 2006 Fundamentals of Business Statistics,35,Least Squares Regression P
27、roperties,The sum of the residuals from the least squares regression line is 0 ( ) The sum of the squared residuals is a minimum (minimized ) The simple regression line always passes through the mean of the y variable and the mean of the x variable The least squares coefficients are unbiased estimat
28、es of 0 and 1,Fall 2006 Fundamentals of Business Statistics,36,YDI 7.6,The growth of children from early childhood through adolescence generally follows a linear pattern. Data on the heights of female Americans during childhood, from four to nine years old, were compiled and the least squares regres
29、sion line was obtained as = 32 + 2.4x where is the predicted height in inches, and x is age in years. Interpret the value of the estimated slope b1 = 2. 4. Would interpretation of the value of the estimated y-intercept, b0 = 32, make sense here? What would you predict the height to be for a female A
30、merican at 8 years old? What would you predict the height to be for a female American at 25 years old? How does the quality of this answer compare to the previous question?,Fall 2006 Fundamentals of Business Statistics,37,The coefficient of determination is the portion of the total variation in the
31、dependent variable that is explained by variation in the independent variableThe coefficient of determination is also called R-squared and is denoted as R2,Coefficient of Determination, R2,Fall 2006 Fundamentals of Business Statistics,38,Coefficient of Determination, R2,(continued),Note: In the sing
32、le independent variable case, the coefficient of determination iswhere:R2 = Coefficient of determinationr = Simple correlation coefficient,Fall 2006 Fundamentals of Business Statistics,39,Examples of Approximate R2 Values,y,x,y,x,y,x,y,x,Fall 2006 Fundamentals of Business Statistics,40,Examples of A
33、pproximate R2 Values,R2 = 0,No linear relationship between x and y: The value of Y does not depend on x. (None of the variation in y is explained by variation in x),y,x,R2 = 0,Fall 2006 Fundamentals of Business Statistics,41,SPSS Output,Fall 2006 Fundamentals of Business Statistics,42,Standard Error
34、 of Estimate,The standard deviation of the variation of observations around the regression line is called the standard error of estimateThe standard error of the regression slope coefficient (b1) is given by sb1,Fall 2006 Fundamentals of Business Statistics,43,SPSS Output,Fall 2006 Fundamentals of B
35、usiness Statistics,44,Comparing Standard Errors,y,y,y,x,x,x,y,x,Variation of observed y values from the regression line,Variation in the slope of regression lines from different possible samples,Fall 2006 Fundamentals of Business Statistics,45,Inference about the Slope: t Test,t test for a populatio
36、n slope Is there a linear relationship between x and y? Null and alternative hypotheses H0: 1 = 0 (no linear relationship) H1: 1 0 (linear relationship does exist) Test statistic,where:b1 = Sample regression slopecoefficient1 = Hypothesized slopesb1 = Estimator of the standarderror of the slope,Fall
37、 2006 Fundamentals of Business Statistics,46,Estimated Regression Equation:,The slope of this model is 0.1098 Does square footage of the house affect its sales price?,Inference about the Slope: t Test,(continued),Fall 2006 Fundamentals of Business Statistics,47,Inferences about the Slope: t Test Exa
38、mple,H0: 1 = 0 HA: 1 0,Test Statistic: t = 3.329,There is sufficient evidence that square footage affects house price,From Excel output:,Reject H0,t,b1,Decision: Conclusion:,Reject H0,Reject H0,a/2=.025,-t/2,Do not reject H0,0,t(1-/2),a/2=.025,-2.3060,2.3060,3.329,d.f. = 10-2 = 8,Fall 2006 Fundament
39、als of Business Statistics,48,Regression Analysis for Description,Confidence Interval Estimate of the Slope:,Excel Printout for House Prices:,At 95% level of confidence, the confidence interval for the slope is (0.0337, 0.1858),d.f. = n - 2,Fall 2006 Fundamentals of Business Statistics,49,Regression
40、 Analysis for Description,Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $185.80 per square foot of house size,This 95% confidence interval does not include 0. Conclusion: There is a significant relationship be
41、tween house price and square feet at the .05 level of significance,Fall 2006 Fundamentals of Business Statistics,50,Residual Analysis,Purposes Examine for linearity assumption Examine for constant variance for all levels of x Evaluate normal distribution assumption Graphical Analysis of Residuals Ca
42、n plot residuals vs. x Can create histogram of residuals to check for normality,Fall 2006 Fundamentals of Business Statistics,51,Residual Analysis for Linearity,Not Linear,Linear,x,residuals,x,y,x,y,x,residuals,Fall 2006 Fundamentals of Business Statistics,52,Residual Analysis for Constant Variance,Non-constant variance,Constant variance,x,x,y,x,x,y,residuals,residuals,Fall 2006 Fundamentals of Business Statistics,53,Residual Output,
