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ANOVA andLinear Regression,[object Object]
Analysis of Variance(ANOVA),[object Object]
BUS B272 Unit 1,[object Object],Analysis of Variance,[object Object],The Analysis of Variance (ANOVA) is a procedure that tests to determine whether differences exist between two or more populations.,[object Object],The techniques analyzes the variance of the data to determine whether we can infer that the populations differ.,[object Object]
	One way (Single-factor) analysis of variance,[object Object],ANOVA assumptions,[object Object],F  test for difference among k  means,[object Object],BUS B272 Unit 1,[object Object],Topics,[object Object]
BUS B272 Unit 1,[object Object],General Experimental Setting,[object Object],Investigator controls one or more independent variables,[object Object],Called treatments or factors,[object Object],Each treatment contains two or more levels (or categories/classifications),[object Object],Observe effects on dependent variable,[object Object],Response to different levels of independent variable,[object Object],Experimental design: the plan used to test hypothesis,[object Object]
BUS B272 Unit 1,[object Object],Completely Randomized Design,[object Object],Experimental units (subjects) are assigned randomly to treatments,[object Object],Subjects are assumed homogeneous,[object Object],Only one factor or independent variable,[object Object],With two or more treatment levels,[object Object],Analyzed by,[object Object],One-way analysis of variance (one-way ANOVA),[object Object]
BUS B272 Unit 1,[object Object],Randomized Design Example,[object Object],,[object Object],,[object Object],,[object Object]
BUS B272 Unit 1,[object Object],One-way Analysis of Variance F  Test,[object Object],Evaluate the difference among the mean responses of 2 or more (k) populations,[object Object],e.g. : Several types of tires, oven 	temperature settings, different types 	of marketing strategies,[object Object]
BUS B272 Unit 1,[object Object],[object Object]
This condition must be met
Populations are normally distributed
F  test is robust to moderate departure from normality
Populations have equal variancesAssumptions of ANOVA,[object Object]
BUS B272 Unit 1,[object Object],Hypotheses of One-Way ANOVA,[object Object],All population means are equal ,[object Object],No treatment effect (no variation in means among groups),[object Object],At least one population mean is different (others may be the same!) ,[object Object],There is treatment effect ,[object Object],Does not mean that all population means are different,[object Object]
BUS B272 Unit 1,[object Object],One-way ANOVA (No Treatment Effect),[object Object],The Null Hypothesis is True,[object Object]
BUS B272 Unit 1,[object Object],One-way ANOVA (Treatment Effect Present),[object Object],The Null Hypothesis is NOT True,[object Object]
BUS B272 Unit 1,[object Object],One-way ANOVA(Partition of Total Variation),[object Object],Total Variation SS(Total),[object Object],Variation Due to Treatment   SST,[object Object],Variation Due to Random Sampling   SSE,[object Object],+,[object Object],=,[object Object]
BUS B272 Unit 1,[object Object],ANOVA set-up,[object Object]
BUS B272 Unit 1,[object Object],Total Variation,[object Object],     : the i-th observation in group j,[object Object],     : the number of observations in group j,[object Object],n   : the total number of observations in all groups,[object Object],k   :  the number of groups,[object Object],the overall or grand mean,[object Object]
BUS B272 Unit 1,[object Object],Total Variation,[object Object],(continued),[object Object]
BUS B272 Unit 1,[object Object],Among-Treatments Variation,[object Object],Variation Due to Differences Among Groups,[object Object]
BUS B272 Unit 1,[object Object],Among-Treatments Variation,[object Object],(continued),[object Object]
BUS B272 Unit 1,[object Object],Summing the variation within each treatment and then adding over all treatments.,[object Object],Within-Treatment Variation,[object Object]
BUS B272 Unit 1,[object Object],Within-Treatment Variation,[object Object],(continued),[object Object]
BUS B272 Unit 1,[object Object],Within-Treatment Variation,[object Object],(continued),[object Object],[object Object]
For 2 groups, use t-test.  F test is more limited.For k = 2, this is the pooled-variance in the t-test.,[object Object]
BUS B272 Unit 1,[object Object],One-way ANOVAF  Test Statistic,[object Object],Test statistic:,[object Object],MST is mean squares among or between variances,[object Object],MSE is mean squares within or error variances,[object Object],Degrees of freedom:,[object Object]
BUS B272 Unit 1,[object Object],One-way ANOVA Summary Table,[object Object]
BUS B272 Unit 1,[object Object],Features of One-way ANOVA F Statistic,[object Object],The F statistic is the ratio of the among estimate of variance and the within estimate of variance.,[object Object],The ratio must always be positive,[object Object], df1 = k -1 will typically be small,[object Object],df2 = n - k  will typically be large,[object Object],The ratio should be closed to 1 if the null is true.,[object Object]
BUS B272 Unit 1,[object Object],One-way ANOVA F  Test Example,[object Object],As production manager, you want to see if three filling machines have different mean filling times.  You assign 15 similarly trained and experienced workers, five per machine, to the machines.  At the 0.05 significance level, is there a difference in mean filling times?,[object Object],Machine1Machine2Machine3	25.40	      23.40	      20.00	26.31	      21.80	      22.20	24.10	      23.50	      19.75	23.74	      22.75	      20.60	25.10	      21.60	      20.40,[object Object]
BUS B272 Unit 1,[object Object],One-way ANOVA Example: Scatter Diagram,[object Object],Machine1Machine2Machine3	25.40	      23.40	      20.00	26.31	      21.80	      22.20	24.10	      23.50	      19.75	23.74	      22.75	      20.60	25.10	      21.60	      20.40,[object Object],Time in Seconds,[object Object],27,[object Object],26,[object Object],25,[object Object],24,[object Object],23,[object Object],22,[object Object],21,[object Object],20,[object Object],19,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object],•,[object Object]
BUS B272 Unit 1,[object Object],Machine 1Machine 2Machine 3	25.40	     23.40	      20.00	26.31	     21.80	      22.20	24.10	     23.50	      19.75	23.74	     22.75	      20.60	25.10	     21.60	      20.40,[object Object],One-way ANOVA Example Computations,[object Object]
BUS B272 Unit 1,[object Object]
BUS B272 Unit 1,[object Object],Summary Table,[object Object],MST/MSE,[object Object],=25.602,[object Object],3-1=2,[object Object],47.1640,[object Object],23.5820,[object Object],15-3=12,[object Object],11.0532,[object Object],0.9211,[object Object],15-1=14,[object Object],58.2172,[object Object]
BUS B272 Unit 1,[object Object], = 0.05,[object Object],F,[object Object],0,[object Object],One-way ANOVA Example Solution,[object Object],Critical Value(s):,[object Object],H0: 1 = 2 = 3,[object Object],H1: Not all the means are equal,[object Object],Test Statistic: ,[object Object],3.89,[object Object],df1= 2      df2 = 12,[object Object],Reject H0 at  = 0.05,[object Object],There is evidence to believe that at least one  i  differs from the rest.,[object Object]
BUS B272 Unit 1,[object Object],Computer Application,[object Object],To obtain the Microsoft Excel computer output in the previous page, first enter the data into c columns in an Excel file, then follow the commands:,[object Object],	Tools/ Data Analysis/ Anova: Single Factor,[object Object]
BUS B272 Unit 1,[object Object],Computer Output using Data Analysis of Excel,[object Object]
Exercise 1,[object Object],The manager of a large department store wants to test if the average size of customer transactions differs with four types of payment: Visa card, company card, cash or cheque. If there are differences in the average customer transaction size among the four types of payment, the manager will further investigate which types of payment will give rise to higher transaction volumes and hence he will design an appropriate promotional programme. A random sample of 54 customer transactions using various types of payment was drawn during the past two months. With reference to sampled data, the sample statistics are obtained as follows:,[object Object],BUS B272 Unit 1,[object Object],Test if differences of average customer transaction size exist among the four types of payment at a 0.05 level of significance.,[object Object]
Exercise 1,[object Object],BUS B272 Unit 1,[object Object],One factor is involved, i.e. the type of payment. Under this factor, there are k = 4 treatments (or factor levels) which represent the four types of payment: Visa card, company card, cash and cheque. The experimental units are customer transactions.,[object Object]
Exercise 1,[object Object],Since the test statistic of 39.16 is greater than the critical value of 2.80, reject H0. At 0.05 level of significance, there is evidence to reveal that the average customer transaction sizes are significantly different among the four types of payment. ,[object Object],BUS B272 Unit 1,[object Object]
Can ANOVA be replaced by t-Test?,[object Object],t-Test : any difference between two population means μ1 and μ2,[object Object],Multiple t-tests are required for more than two population means,[object Object],Conducting multiple tests increases the probability of making Type I errors. ,[object Object],	E.g. compare 6 population means, if use ANOVA with significant level 5%, there will be a 5% chance we reject the null hypothesis when it is true. ,[object Object],	If we use t-test, we need to perform 15 tests and if same 5% significant level is set, the chance of a Type I error will be,[object Object],		1 – (1 - 0.05)15 = 0.54,[object Object],BUS B272 Unit 1,[object Object]
Linear Regression,[object Object]
BUS B272 Unit 1,[object Object],Linear Regression,[object Object],Origin of regression,[object Object],Determining the simple linear regression equation,[object Object],Assessing the fitness of the model ,[object Object],Correlation analysis,[object Object],Estimation and prediction ,[object Object],Assumptions of regression and correlation,[object Object]
BUS B272 Unit 1,[object Object],Origin of Regression,[object Object],“Regression," from a Latin root meaning "going back," is a series of statistical methods used in studying the relationship between two variables and were first employed by Francis Galton in 1877. ,[object Object],Galton was interested in studying the relationship between a father’s height and the son’ s height. Making use of the “regression” method, he found that son’s height regress to the overall mean and the method is then called “regression”.,[object Object]
BUS B272 Unit 1,[object Object],Linear Regression Analysis,[object Object],Linear Regression analysis is used primarily to model and describe linear relationship and provide prediction among variables ,[object Object],Predicts the value of a dependent (response) variable based on the value of at least one independent (explanatory) variable,[object Object],Express statistically the effect of the independent variables on the dependent variable,[object Object]
BUS B272 Unit 1,[object Object],Types of Regression Models,[object Object],Positive Linear Relationship,[object Object],Relationship NOT Linear,[object Object],Negative Linear Relationship,[object Object],No Relationship,[object Object]
BUS B272 Unit 1,[object Object],Simple Linear Regression Model,[object Object],The relationship between two variables, sayX and Y,  is described by a linear function.,[object Object],The change of the variable Y, (called dependent or response variable) is associated with the change in the other variable X(called independent or explanatory variable). ,[object Object],Explore the dependency of Y on X.,[object Object]
(4, 5),[object Object],(2, 2.5),[object Object],(3, 2.5),[object Object],(1, 2),[object Object],Why Regression?,[object Object],The larger the sum of squares, the poor the estimate.,[object Object],X,[object Object],1,[object Object],2,[object Object],3,[object Object],4,[object Object],Y,[object Object],2,[object Object],2.5,[object Object],2.5,[object Object],5,[object Object],BUS B272 Unit 1,[object Object]
BUS B272 Unit 1,[object Object],Linear Relationship,[object Object],We wish to study whether there is any association between two quantitative variables, sayX and Y,[object Object],If ‘Y tends to increase as X increases’ ,[object Object],If ‘Y tends to decrease as X increases’,[object Object],	If the corresponding magnitude of increase or decrease follows a specific proportion, the relationship identified is said to be a linear one.,[object Object],–  apositive relationship,[object Object],–  anegative relationship,[object Object]
BUS B272 Unit 1,[object Object],Scatter Diagram,[object Object],A scatter diagram is a graph plotted for all X-Y pairs of the sample data.,[object Object],By viewing a scatter diagram, one can determine whether a relationship exists between the two variables. It can also suggest the likely mathematical form of that relationship that allow one to judge initially and intuitively whether or not there exists a linear relationship between the two variables involved.,[object Object]
BUS B272 Unit 1,[object Object],Example,[object Object],The level of air pollution at Kwun Tong and the total number of consultations relating to respiratory diseases in a public clinic in the area were recorded during a specific time period on 14 randomly selected days.,[object Object]
BUS B272 Unit 1,[object Object],Population Linear Regression,[object Object],Population regression line is a straight line that describes the dependence of the average value (conditional mean) of one variable on the other ,[object Object],Random Error,[object Object],Population SlopeCoefficient ,[object Object],Population Y  intercept ,[object Object],Dependent (Response) Variable,[object Object],PopulationRegression,[object Object],Line  ,[object Object],(conditional mean),[object Object],Independent (Explanatory) Variable,[object Object]
BUS B272 Unit 1,[object Object],Population Linear Regression,[object Object],(continued),[object Object],Random Error (vertical discrepancies or residual for point i ),[object Object],Y,[object Object],(Observed Value of Y) =,[object Object],(Conditional Mean),[object Object],X,[object Object],Observed Value of Y,[object Object]
BUS B272 Unit 1,[object Object],Least Squares Method,[object Object],The line fitted by least squares is the one that makes the sum of squares of all those vertical discrepancies (residuals) as small as possible, i.e. minimum of ,[object Object],which is the sum of squared residuals.,[object Object]
BUS B272 Unit 1,[object Object],Sample Y  intercept,[object Object],Residual,[object Object],Sample regression line is formed by the point estimates of     and     , i.e.,     and    .  It provides an estimate of the population regression line as well as a predicted value of Y,[object Object],Sample Linear Regression,[object Object],Samplecoefficient of slope,[object Object],Sample regression line ,[object Object],(Fitted regression line or predicted value),[object Object]
BUS B272 Unit 1,[object Object],Sample Linear Regression,[object Object],(continued),[object Object],and      are obtained by finding the specific values of       and      that minimizes the sum of the squared residuals,[object Object]
BUS B272 Unit 1,[object Object],Coefficients of Sample Linear Regression,[object Object],For ,[object Object]
BUS B272 Unit 1,[object Object],Interpretation of the Slope and the Intercept,[object Object],is the average value of Y when the value of X  is zero.,[object Object],		 measures the change in the average value of Y as a result of a one-unit change in X.,[object Object]
BUS B272 Unit 1,[object Object],(continued),[object Object],is the estimated average value of Y when the value of X  is zero.,[object Object],	  	 is the estimated change in the average value of Y as a result of one-unit change in X.,[object Object],Interpretation of the Slope and the Intercept,[object Object]
BUS B272 Unit 1,[object Object],Example 1 : Simple Linear Regression,[object Object],Suppose that you want to examine the linear dependency of the annual sales among seven stores on their size in square footage. Sample data for seven stores were obtained. Find the equation of the straight line that fits the data best.,[object Object],Annual Store	   Square 	 Sales		     Feet	($1000),[object Object],   1           1,726	  3,681,[object Object],   2           1,542	  3,395,[object Object],   3	     2,816	  6,653,[object Object],   4	     5,555	  9,543,[object Object],   5	     1,292	  3,318,[object Object],   6	     2,208	  5,563,[object Object],   7	     1,313	  3,760	,[object Object]
BUS B272 Unit 1,[object Object],Example 1 : Scatter Diagram,[object Object],Excel Output,[object Object]
BUS B272 Unit 1,[object Object],Computation of Regression Coefficient,[object Object],	                     Annual ,[object Object],          Square     Sales,[object Object],Store   Feet      ($1000) ,[object Object],XY,[object Object],   1       1,726      3,681	  	,[object Object],   2       1,542      3,395	  	,[object Object],   3	  2,816      6,653	,[object Object],   4	  5,555      9,543	,[object Object],   5	  1,292      3,318	  	,[object Object],   6	  2,208      5,563	,[object Object],   7	  1,313      3,760	  	,[object Object], 2,979,076,[object Object], 2,377,764,[object Object], 7,929,856,[object Object],30,858,025,[object Object], 1,669,264,[object Object], 4,875,264,[object Object], 1,723,969,[object Object],13,549,761,[object Object],11,526,025,[object Object],44,262,409,[object Object],91,068,849,[object Object],11,009,124,[object Object],30,946,969,[object Object],14,137,600,[object Object],  6,353,406 ,[object Object],  5,235,090,[object Object],18,734,848,[object Object],53,011,365,[object Object],  4,286,856,[object Object],12,283,104,[object Object],  4,936,880,[object Object],16,452,[object Object],35,913,[object Object],104,841,549,[object Object],52,413,218,[object Object],216,500,737,[object Object]
BUS B272 Unit 1,[object Object],Computation of Regression Coefficient,[object Object]
BUS B272 Unit 1,[object Object],Example 1 : Equation for the Sample 	Regression Line,[object Object],Yi = 1636.415 +1.487Xi,[object Object],,[object Object]
BUS B272 Unit 1,[object Object],Example 1 : Interpretation of Results ,[object Object],The slope of 1.487 means that for each increase of one unit in X, we predict the average of Y to increase by an estimated 1.487 units.,[object Object],The model estimates that for each increase of one square foot in the size of the store, the expected annual sales are predicted to increase by $1487.,[object Object]
BUS B272 Unit 1,[object Object],Predicting Annual Sales Based on Square Footage ,[object Object],Suppose that we would like to use the fitted model to predict the average annual sales for a store with 4,000 square feet.,[object Object]
BUS B272 Unit 1,[object Object],Interpolation versus Extrapolation,[object Object],For using regression line for prediction purpose, it is not appropriate to make predictions beyond the relevant range (in the previous example: (1,292, 5,555)) of the independent variable.,[object Object],That is, we may interpolate within the relevant range of X  values, but we SHOULD NOT extrapolate beyond the range of X values. For example, it is not appropriate to predict the average annual sales for a store with 7,000 square feet since it is beyond the range of X  values, i.e., (1,292, 5,555).,[object Object]
BUS B272 Unit 1,[object Object],Causal Relationship?,[object Object],In general, when there is a relationship identified between X and Y using regression analysis, we usually would say that ‘X is associated with Y’ instead of saying ‘X causes Y’.,[object Object],We cannot claim that two variables are related by cause and effect just because there is a statistical relationship between the two. In fact, you cannot infer a causal relationship from statistics alone. ,[object Object]
BUS B272 Unit 1,[object Object],For example, the price of dog food and houses, may well be positively correlated over time. ,[object Object],When you collect data concerning the price of dog food and the price of houses over time, you might end up with an inference that they have a positive relationship, but can you conclude that an increase in the price of dog food would directly cause the price of houses to increase too? ,[object Object],It might be that an inflationary force is influencing both and hence they can be seen to move in the same general direction over time. ,[object Object]
BUS B272 Unit 1,[object Object],Computer Application,[object Object],Import the data into two adjacent columns in an Excel file and then click Tools/Data Analysis/ Regression(See page 624-5 for detail description).,[object Object]
BUS B272 Unit 1,[object Object],Example 1: Computer Output,[object Object]
BUS B272 Unit 1,[object Object],Exercise 2,[object Object],Consider the example about the level of air pollution at Kwun Tong and the total number of consultations that relate to respiratory diseases in a public clinic in the area. The corresponding data were given as follows:,[object Object]
BUS B272 Unit 1,[object Object],Exercise 1,[object Object],(a)	Determine the sample regression line to predict 	the number of consultations by the level of 	pollution.,[object Object],(b)	Interpret the coefficients.,[object Object],Solution:,[object Object]
BUS B272 Unit 1,[object Object],Exercise 1,[object Object],For      , each additional increase in pollution level, the number of consultations increases, on average by 0.456701074. ,[object Object],No meaningful interpretation for       can be made, as the range of x does not include zero.,[object Object]
BUS B272 Unit 1,[object Object],Assessing the simple linear regression model,[object Object],From time to time, after we have set up a linear regression model, we wish to assess the fitness of the model. That is, we wish to find out how well the model fit to the given data. For a good fit, the data as a whole should be quite close to the regression line and the independent variable can thus be used to predict the value of the dependent variable with high accuracy. ,[object Object],To examine how well the independent variable predicts the dependent variable, we need to develop several measures of variation.,[object Object]
BUS B272 Unit 1,[object Object],Total Sample Variability,[object Object],Unexplained Variability,[object Object],=,[object Object],Explained Variability,[object Object],+,[object Object],Measure of Variation: The Sum of Squares,[object Object],SS(Total)         =SSR            +           SSE,[object Object]
BUS B272 Unit 1,[object Object],Measure of Variation: The Sum of Squares,[object Object],SS(Total) = total sum of squares ,[object Object],Measures the variation of the Yi values around their mean Y,[object Object],SSR = regression sum of squares ,[object Object],Explained variation attributable to the relationship between X and Y,[object Object],SSE = error sum of squares ,[object Object],Variation attributable to factors other than the relationship between X and Y  (Unexplained variation),[object Object],(continued),[object Object]
BUS B272 Unit 1,[object Object],Measure of Variation: The Sum of Squares,[object Object],_,[object Object],SS(Total) = (Yi  – Y )2,[object Object],(continued),[object Object],Y,[object Object],Yi,[object Object],,[object Object],SSE=(Yi - Yi)2,[object Object],_,[object Object],,[object Object],_,[object Object],SSR = (Yi - Y)2,[object Object],_,[object Object],Y,[object Object],X,[object Object],Xi,[object Object]
BUS B272 Unit 1,[object Object]
BUS B272 Unit 1,[object Object],Standard Error of Estimate,[object Object],The standard deviation of the variation of observations around the regression line. ,[object Object]
The smallest value that        can assume is 0, which occurs when SSE = 0, that is, when all the points fall on the regression line. Thus, when      is small, the fit is excellent, and the linear regression model is likely to be an effective analytical and forecasting tool.,[object Object],When      is large, the regression model is a poor one, it is of little value to be used.,[object Object],BUS B272 Unit 1,[object Object],Standard Error of Estimate,[object Object]
BUS B272 Unit 1,[object Object],The Coefficient of Determination (r 2  or R 2 ),[object Object],By themselves, SSR, SSE  and SS(Total) provide little that can be directly interpreted.  A simple ratio of SSR and SS(Total) provides a measure of the usefulness of the regression equation.,[object Object],Measures the proportion of variation in Y  that is explained by the independent variable X  in the regression model ,[object Object]
BUS B272 Unit 1,[object Object],Coefficients of Determination (r 2),[object Object],r2 = 1,[object Object],Y,[object Object],Y,[object Object],r2 = 1,[object Object],^,[object Object],Y,[object Object], = ,[object Object],b,[object Object], + ,[object Object],b,[object Object],X,[object Object],i,[object Object],0,[object Object],1,[object Object],i,[object Object],^,[object Object],Y,[object Object], = ,[object Object],b,[object Object], + ,[object Object],b,[object Object],X,[object Object],i,[object Object],0,[object Object],1,[object Object],i,[object Object],X,[object Object],X,[object Object],r2 = 0,[object Object],r2 = 0.8,[object Object],Y,[object Object],Y,[object Object],^,[object Object],^,[object Object],Y,[object Object], = ,[object Object],b,[object Object], + ,[object Object],b,[object Object],X,[object Object],Y,[object Object], = ,[object Object],b,[object Object], + ,[object Object],b,[object Object],X,[object Object],i,[object Object],0,[object Object],1,[object Object],i,[object Object],i,[object Object],0,[object Object],1,[object Object],i,[object Object],X,[object Object],X,[object Object]
BUS B272 Unit 1,[object Object],Coefficient of Correlation,[object Object],Coefficient of correlation is used to measure strength of association (linear relationship) between two numerical variables),[object Object],Only concerned with strength of the relationship,[object Object],No causal effect is implied,[object Object]
BUS B272 Unit 1,[object Object],(continued),[object Object],Population correlation coefficient is denoted by  (Rho).,[object Object],Sample correlation coefficient is denoted by r . It is an estimate of   and is used to measure the strength of the linear relationship in the sample observations.,[object Object],Coefficient of Correlation,[object Object]
BUS B272 Unit 1,[object Object],Coefficient of Correlation,[object Object]
BUS B272 Unit 1,[object Object],Sample of Observations from Various r  Values,[object Object],Y,[object Object],Y,[object Object],Y,[object Object],X,[object Object],X,[object Object],X,[object Object],r = –1,[object Object],r = –0.6,[object Object],r = 0,[object Object],Y,[object Object],Y,[object Object],X,[object Object],X,[object Object],r = 0.6,[object Object],r = 1,[object Object]
BUS B272 Unit 1,[object Object],Features of r and r,[object Object],Unit free,[object Object],Range between –1 and 1,[object Object],The closer to –1, the stronger the negative linear relationship,[object Object],The closer to 1, the stronger the positive linear relationship,[object Object],The closer to 0, the weaker the linear relationship,[object Object]
BUS B272 Unit 1,[object Object],There is also a more systematic way to assess model fitness, i.e., to perform a hypothesis testing on the slope of the regression line.,[object Object],Inference about the Slope,[object Object],If the two variables involved are not at all linearly related, one could observe from the scatter diagram shown on the right that the slope of the regression line will be zero.,[object Object]
BUS B272 Unit 1,[object Object],Hence, we can determine whether a significant relationship between the variables X  and Y exists by testing whether 	(the true slope) is equal to zero.,[object Object],Inference about the Slope,[object Object],(There is no linear relationship),[object Object],(There is a linear relationship),[object Object],If       is rejected, there is evidence to believe that a linear relationship exists between X  and Y.,[object Object]
BUS B272 Unit 1,[object Object],The standard error of the slope,[object Object],The estimated standard error of     . ,[object Object]
BUS B272 Unit 1,[object Object],Inference about the Slope: t  Test,[object Object],t  test for a population slope,[object Object],Is there a linear dependency of Y on X ?,[object Object],Null and alternative hypotheses,[object Object],H0:  1 = 0	(no linear dependency),[object Object],H1:  1 0	(linear dependency),[object Object],Test statistic:,[object Object]
BUS B272 Unit 1,[object Object],Example: Store Sales,[object Object],Data for Seven Stores:,[object Object],Estimated Regression Equation:,[object Object],Annual Store	   Square 	 Sales		     Feet	($000),[object Object],   1           1,726	  3,681,[object Object],   2           1,542	  3,395,[object Object],   3	     2,816	  6,653,[object Object],   4	     5,555	  9,543,[object Object],   5	     1,292	  3,318,[object Object],   6	     2,208	  5,563,[object Object],   7	     1,313	  3,760	,[object Object],,[object Object],Yi = 1636.415 +1.487Xi,[object Object],The slope of this model is 1.487. ,[object Object],Is square footage of the store affecting its annual sales?,[object Object]
H0: 1 = 0          0.05,[object Object],H1: 1 0          df7 - 2  = 5,[object Object],Test Statistic: ,[object Object],BUS B272 Unit 1,[object Object]
BUS B272 Unit 1,[object Object],Inferences about the Slope: t  Test Example,[object Object],Reject,[object Object],Reject,[object Object],0.025,[object Object],0.025,[object Object],0,[object Object],2.5706,[object Object],-2.5706,[object Object],Decision:,[object Object],Conclusion:,[object Object],Critical Value(s):,[object Object],Reject H0,[object Object],At 5% level of significance, there is evidence to reveal that square footage is associated with annual sales.,[object Object]
BUS B272 Unit 1,[object Object],(No linear relationship),[object Object],(A linear relationship),[object Object],(No positive linear relationship),[object Object],(A positive linear relationship),[object Object],(No negative linear relationship),[object Object],(A negative linear relationship),[object Object],Inferences about the Slope,[object Object]
BUS B272 Unit 1,[object Object],Exercise 3,[object Object],	Consider the data of Exercise 2 about the level of air pollution at Kwun Tong and the total number of consultations that relate to respiratory diseases in a public clinic in the area. ,[object Object],Test at the 5% level of significance to determine whether level of air pollution and the total number of consultations are positively linearly related.,[object Object]
BUS B272 Unit 1,[object Object],Solution:,[object Object],0.05;   df14 - 2  = 12,[object Object]
BUS B272 Unit 1,[object Object],Exercise 3,[object Object]
BUS B272 Unit 1,[object Object],Computer Output,[object Object],For two-tailed test,[object Object]
BUS B272 Unit 1,[object Object],Exercise 3,[object Object],Decision:,[object Object],Conclusion:,[object Object],Reject H0,[object Object],Critical Value(s):,[object Object],Reject H0,[object Object],At 5% level of significance, there is evidence to believe that level of air pollution and total number of consultations are positively linearly related.,[object Object],0.05,[object Object],0,[object Object],1.7823,[object Object]
BUS B272 Unit 1,[object Object],You have seen how can we assess the model fitness. If the model fits satisfactorily, we can use it to forecast and estimate values of the dependent variable. ,[object Object],We can obtain a point prediction of Y with a given value of X  using the linear regression line.,[object Object],Confidence interval about the particular value of Y  or the average of Y  for a given value of X  can also be computed if desired.,[object Object],Estimation of Mean Values,[object Object]
BUS B272 Unit 1,[object Object],Estimation of Mean Values,[object Object],Confidence interval estimate for             :,[object Object],The mean of Y given a particular  ,[object Object],Size of interval varies according to distance away from mean,    ,[object Object],Standard error of the estimate,[object Object],t value from table with df = n - 2,[object Object]
BUS B272 Unit 1,[object Object],Prediction of Individual Values,[object Object],Prediction interval for individual response Yi at a particular ,[object Object],Addition of one increases width of interval from that for the mean of Y,[object Object]
BUS B272 Unit 1,[object Object],Interval Estimates for Different Values of X,[object Object],Confidence Interval for the mean of Y,[object Object],Prediction Interval for a individual Yi,[object Object],Y,[object Object],,[object Object],Yi = b0 + b1Xi,[object Object],X,[object Object],Y given X,[object Object]
BUS B272 Unit 1,[object Object],Example: Stores Sales,[object Object],Data for seven stores:,[object Object],Predict the annual sales for a store with 2000 square feet.,[object Object],Annual Store	   Square 	 Sales		     Feet	($000),[object Object],   1           1,726	  3,681,[object Object],   2           1,542	  3,395,[object Object],   3	     2,816	  6,653,[object Object],   4	     5,555	  9,543,[object Object],   5	     1,292	  3,318,[object Object],   6	     2,208	  5,563,[object Object],   7	     1,313	  3,760	,[object Object],Regression Model Obtained:,[object Object],,[object Object],Yi = 1636.415 +1.487Xi,[object Object]
Estimation of Mean Values: Example,[object Object],Confidence Interval Estimate for,[object Object],Find the 95% confidence interval for the average annual sales for a 2,000 square-foot store.,[object Object],,[object Object],Predicted Sales Yi = 1636.415 +1.487Xi = 4609.68 ($000),[object Object],tn-2 = t5 = 2.571,[object Object],X = 2350.29,[object Object],BUS B272 Unit 1,[object Object]
Prediction Interval for Y : Example,[object Object],Prediction Interval for Individual Y,[object Object],Find the 95% prediction interval                                           for the annual sales of a 2,000 square-foot store,[object Object],,[object Object],Predicted Sales Yi = 1636.415 +1.487Xi = 4609.68 ($000),[object Object],tn-2 = t5 = 2.571,[object Object],X = 2350.29,[object Object],BUS B272 Unit 1,[object Object]
BUS B272 Unit 1,[object Object],Computer Application,[object Object],Commands:Tools/ Data Analysis Plus/ Prediction Interval.,[object Object]
BUS B272 Unit 1,[object Object],Computer Output,[object Object]
BUS B272 Unit 1,[object Object],Linear Regression Assumptions,[object Object],1.  Normality,[object Object],Y values are normally distributed for each X,[object Object],Probability distribution of error is normal,[object Object],2.	Homoscedasticity (Constant Variance),[object Object],3.	Independence of Errors,[object Object]
BUS B272 Unit 1,[object Object],[object Object]
 For each X value, the “spread” or variance around the regression line is the same.Variation of Errors around the Regression Line,[object Object],f(e),[object Object],Y,[object Object],X2,[object Object],X1,[object Object],X,[object Object],Sample Regression Line,[object Object],.,[object Object]
Multiple Regression,[object Object]
BUS B272 Unit 1,[object Object],Introduction,[object Object],Extension of the simple linear regression model to allow for any fixed number of independent variables. That is, the number of independent variables could be more than one.,[object Object]
BUS B272 Unit 1,[object Object],Multiple Linear Regression,[object Object],To make use of computer printout to ,[object Object],Assess the model,[object Object],How well it fits the data,[object Object],Is it useful,[object Object],Are any required conditions violated?,[object Object],Employ the model,[object Object],Interpreting the coefficients,[object Object],Predictions using the prediction equation,[object Object],Estimating the expected value of the dependent variable,[object Object]
BUS B272 Unit 1,[object Object],Allow for k independent variables to potentially be related to the dependent variable,[object Object],y = b0 + b1x1+ b2x2 + …+ bkxk + e,[object Object],Regression,[object Object],Coefficients,[object Object],Random error ,[object Object],variable,[object Object],Dependent variable,[object Object],Independent variables,[object Object],Model and Required Conditions,[object Object]
Multiple Regression for k = 2, Graphical Demonstration,[object Object],X,[object Object],1,[object Object],The simple linear regression model,[object Object],allows for one independent variable, “x”,[object Object],for y = b0 + b1x + e,[object Object],y,[object Object],y = b0 + b1x1 + b2x2,[object Object],y = b0 + b1x1 + b2x2,[object Object],y = b0 + b1x1 + b2x2,[object Object],y = b0 + b1x1 + b2x2,[object Object],y = b0 + b1x1 + b2x2,[object Object],y = b0 + b1x1 + b2x2,[object Object],y = b0 + b1x1 + b2x2,[object Object],The multiple linear regression model,[object Object],allows for more than one independent variable.,[object Object],Y = b0 + b1x1 + b2x2  + e,[object Object],X2,[object Object],BUS B272 Unit 1,[object Object]
BUS B272 Unit 1,[object Object],The errore is normally distributed.,[object Object],The mean is equal to zero and the standard deviation is constant (se)for all values of y. ,[object Object],The errors are independent.,[object Object],Required conditions for the error variable,[object Object]
BUS B272 Unit 1,[object Object],Estimating the Coefficients andAssessing the Model,[object Object],The procedure used to perform multiple regression analysis:,[object Object],[object Object]
Assess the model fitness using statistics obtained from the sample.
If the model assessment indicates good fit to the data, use it to interpret the coefficients and generate predictions.,[object Object]
Estimating the Coefficients and Assessing the Model, Example,[object Object],Physical,[object Object],Profitability,[object Object],Margin (%),[object Object],Market ,[object Object],awareness,[object Object],Competition,[object Object],Customers,[object Object],Community,[object Object],Number,[object Object],Office,[object Object],space,[object Object],Income,[object Object],Distance,[object Object],Nearest,[object Object],Enrollment,[object Object],Median,[object Object],household,[object Object],income ,[object Object],of nearby,[object Object],area (in $thousands),[object Object],Number of ,[object Object],hotels/motels,[object Object],rooms within ,[object Object],3 miles from ,[object Object],the site,[object Object],Enrollemnt in nearby university or college (in thousands),[object Object],Distance to,[object Object], the downtown,[object Object],core (in miles),[object Object],Number of miles to closest competition,[object Object],Office space in nearby community,[object Object],BUS B272 Unit 1,[object Object]
BUS B272 Unit 1,[object Object],Estimating the Coefficients and Assessing the Model, Example,[object Object],Data were collected from randomly selected 100 inns that belong to La Quinta, and ran for the following suggested model:,[object Object],Margin = b0 + b1Rooms + b2Nearest + b3Office + 	b4College + b5Income + b6Disttwn,[object Object],Xm18-01,[object Object]
BUS B272 Unit 1,[object Object],Regression Analysis, Excel Output,[object Object],Margin = 38.14 - 0.0076Number +1.65Nearest,[object Object],+ 0.020Office Space +0.21Enrollment,[object Object],+ 0.41Income - 0.23Distance,[object Object],This is the sample regression equation ,[object Object],(sometimes called the prediction equation),[object Object]
BUS B272 Unit 1,[object Object],Model Assessment,[object Object],The model is assessed using two tools:,[object Object],The coefficient of determination,[object Object],The F -test of the analysis of variance,[object Object],The standard error of estimates participates in building the above tools.,[object Object]
BUS B272 Unit 1,[object Object],Standard Error of Estimate,[object Object],The standard deviation of the error is estimated by the Standard Error of Estimate:,[object Object],The magnitude of seis judged by comparing it to ,[object Object]
BUS B272 Unit 1,[object Object],From the printout, se = 5.51 ,[object Object],Calculating the mean value of y, we have,[object Object],It seems se is not particularly small. ,[object Object],Question:Can we conclude the model does not fit the data well? ,[object Object],Standard Error of Estimate,[object Object]
BUS B272 Unit 1,[object Object],Coefficient of Determination,[object Object],The definition is:,[object Object],From the printout,  r 2 = 0.5251,[object Object],52.51% of the variation in operating margin is explained by the six independent variables. 47.49% remains unexplained.,[object Object]
BUS B272 Unit 1,[object Object],Testing the Validity of the Model,[object Object],For testing the validity of the model, the following question is asked:,[object Object],	Is there at least one independent variable linearly related to the dependent variable? ,[object Object],To answer the question we test the hypothesis,[object Object],H0: b1 = b2 = … = bk = 0,[object Object],H1: At least one bi is not equal to zero.,[object Object],If at least one bi is not equal to zero, the model has some validity or usefulness. ,[object Object]
BUS B272 Unit 1,[object Object],Testing the Validity of the La Quinta Inns Regression Model,[object Object],The hypotheses are tested by an ANOVA procedure ( the Excel output),[object Object],MSR / MSE,[object Object],k      =,[object Object],n–k–1 =,[object Object],   n-1  = ,[object Object],SSR,[object Object],MSR=SSR / k,[object Object],SSE,[object Object],MSE=SSE / (n-k-1),[object Object]
BUS B272 Unit 1,[object Object],Testing the Validity of the La Quinta Inns Regression Model,[object Object],	[Total variation in y] SS(Total) = SSR + SSE. ,[object Object],	Large F  results from a large SSR. That implies much of the variation in y can be explained by the regression model; the model is useful, and thus, the null hypothesis should be rejected.  Therefore, the rejection region is:,[object Object],F > Fa, k, n – k – 1,[object Object],while the test statistic is:,[object Object]
BUS B272 Unit 1,[object Object],Testing the Validity of the La Quinta Inns Regression Model,[object Object],Fa, k, n-k-1 = F0.05,6,100-6 -1 = 2.17,[object Object],F = 17.14 > 2.17,[object Object],Conclusion:  There is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.  At least one of the bi is not equal to zero. Thus, at least one independent variable is linearly related to y.   This linear regression model is valid.,[object Object],Also, the p-value (Significance F) = 0.0000; Reject the null hypothesis. ,[object Object]

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