5 follows a Student’s t Distribution with (n-2) degrees of freedom Correlation AnalysisThe null hypothesis of no linear association:where the random variable:follows a Student’s t Distribution with (n-2) degrees of freedom
6 Tests for Zero Population Correlation Let r be the sample correlation coefficient, calculated from a random sample of n pairs of observation from a joint normal distribution. The following tests of the null hypothesishave a significance value :1. To test H0 against the alternativethe decision rule is
7 Tests for Zero Population Correlation (continued) 2. To test H0 against the alternativethe decision rule is
8 Tests for Zero Population Correlation (continued) 3. To test H0 against the two-sided alternativethe decision rule isHere, t n-2, is the number for whichWhere the random variable tn-2 follows a Student’s t distribution with (n – 2) degrees of freedom.
11 Linear Regression Model LINEAR REGRESSION POPULATION EQUATION MODELWhere 0 and 1 are the population model coefficients and is a random error term.
12 Linear Regression Outcomes Linear regression provides two important results:Predicted values of the dependent or endogenous variable as a function of an independent or exogenous variable.Estimated marginal change in the endogenous variable that results from a one unit change in the independent or exogenous variable.
13 Least Squares Procedure The Least-squares procedure obtains estimates of the linear equation coefficients b0 and b1, in the modelby minimizing the sum of the squared residuals eiThis results in a procedure stated asChoose b0 and b1 so that the quantityis minimized. We use differential calculus to obtain the coefficient estimators that minimize SSE..
14 Least-Squares Derived Coefficient Estimators The slope coefficient estimator isAnd the constant or intercept indicator isWe also note that the regression line always goes through the mean X, Y.
15 Standard Assumptions for the Linear Regression Model The following assumptions are used to make inferences about the population linear model by using the estimated coefficients:The x’s are fixed numbers, or they are realizations of random variable, X that are independent of the error terms, i’s. In the latter case, inference is carried out conditionally on the observed values of the x’s.The error terms are random variables with mean 0 and the same variance, 2. The later is called homoscedasticity or uniform variance.The random error terms, I, are not correlated with one another, so that
16 Regression Analysis for Retail Sales Analysis (Figure 10.5) The regression equation isY Retail Sales = X Incomeb0b1
17 Analysis of VarianceThe total variability in a regression analysis, SST, can be partitioned into a component explained by the regression, SSR, and a component due to unexplained error, SSEWith the components defined as,Total sum of squaresError sum of squaresRegression sum of squares
18 Regression Analysis for Retail Sales Analysis (Figure 10.7) The regression equation isY Retail Sales = X Income
19 Coefficient of Determination, R2 The Coefficient of Determination for a regression equation is defined asThis quantity varies from 0 to 1 and higher values indicate a better regression. Caution should be used in making general interpretations of R2 because a high value can result from either a small SSE or a large SST or both.
20 Correlation and R2The multiple coefficient of determination, R2, for a simple regression is equal to the simple correlation squared:
21 Estimation of Model Error Variance The quantity SSE is a measure of the total squared deviation about the estimated regression line, and ei is the residual. An estimator for the variance of the population model error isDivision by n – 2 instead of n – 1 results because the simple regression model uses two estimated parameters, b0 and b1, instead of one.
22 Sampling Distribution of the Least Squares Coefficient Estimator If the standard least squares assumptions hold, then b1 is an unbiased estimator of 1 and has a population varianceand an unbiased sample variance estimator
23 Basis for Inference About the Population Regression Slope Let 1 be a population regression slope and b1 its least squares estimate based on n pairs of sample observations. Then, if the standard regression assumptions hold and it can also be assumed that the errors i are normally distributed, the random variableis distributed as Student’s t with (n – 2) degrees of freedom. In addition the central limit theorem enables us to conclude that this result is approximately valid for a wide range of non-normal distributions and large sample sizes, n.
24 Excel Output for Retail Sales Model (Figure 10.9) The regression equation isY Retail Sales = X IncomeseSSRSSESSTMSRMSEb0b1sb1tb1
25 Tests of the Population Regression Slope If the regression errors i are normally distributed and the standard least squares assumptions hold (or if the distribution of b1 is approximately normal), the following tests have significance value :To test either null hypothesisagainst the alternativethe decision rule is
26 Tests of the Population Regression Slope (continued) 2. To test either null hypothesisagainst the alternativethe decision rule is
27 Tests of the Population Regression Slope (continued) 3. To test the null hypothesisAgainst the two-sided alternativethe decision rule is
28 Confidence Intervals for the Population Regression Slope 1 If the regression errors i , are normally distributed and the standard regression assumptions hold, a 100(1 - )% confidence interval for the population regression slope 1 is given byWhere t(n – 2, /2) is the number for whichAnd the random variable t(n – 2) follows a Student’s t distribution with (n – 2) degrees of freedom.
29 F test for Simple Regression Coefficient We can test the hypothesisagainst the alternativeBy using the F statisticThe decision rule isWe can also show that the F statistic isFor any simple regression analysis.
30 Key Words Analysis of Variance Assumptions for the Least Squares Coefficient EstimatorsBasis for Inference About the Population Regression SlopeCoefficient of Determination, R2Confidence Intervals for PredictionsConfidence Intervals for the Population Regression Slope b1Correlation and R2Estimation of Model Error VarianceF test for Simple Regression CoefficientLeast-Squares ProcedureLinear Regression Outcomes
31 Key Words (continued) Linear Regression Population Equation Model Population ModelSampling Distribution of the Least Squares Coefficient EstimatorTests for Zero Population CorrelationTests of the Population Regression Slope