TEST HYPOTHESIS OF SLOPE COEFFICIENT EQUAL TO VALUE OTHER THANįor non-zero hypthesized value of the slope parameter we need to Further refinement is needed depending on the direction of the.For the t-statistic approach the reported t-statistic isīut the critical value is now t_.05(3) = TINV(0.10,3) = 2.353.For the p-value approach the reported p-value is for a two-sidedĪnd needs to be halved for a one-sided test: p = 0.0405/2 = 0.202.If instead one-sided tests are performed, we need to adjust the TEST HYPOTHESIS OF ZERO SLOPE COEFFICIENT ("TEST OF STATISTICALĮxcel automatically gives output to make this test easy. Other confidence intervals can be obtained.įor example, to find 99% confidence intervals: in the Regression dialogĬheck the Confidence Level box and set the level to 99%. Standard error of the regression is 0.365148.ĩ5% confidence interval for slope coefficient β 2 is from A measure of the fit of the model is R 2 = 0.8.(intercept and x) so in inference we use T(5-2) =T(3). The 95% confidence interval for β 2 is (0.0325,.The slope coefficient has p-value of 0.0405.The slope coefficient has estimated standard error of 0.115.The columns "Lower 95%" and "Upper 95%" values define a 95%Ī simple summary of the above output is that Note that this P-value is for a 2-sided test.įor a 1-sided test divide this P-value by 2 (also checking the sign With n-k degreres of freedom and t-Stat is the computed value of the This equals the Prwhere T is a T-distributed random The column "P-value" gives for hh size are for H0: β 2 = 0 To a T distribution with (n-k) degrees of freedom where here n = 5and k This is the coefficient divided by the standard error: here 0.4 / The second row of the column "t Stat" gives the computed t-statistic Standard deviation) of the least squares estimate of β 1 and β 2 The column "Standard error" gives the standard errors (i.e.the The column "Coefficient" gives the least squares estimates of β 1 Similar interpretation is given for inference on β 1, using Here we focus on inference on β 2, using the row that Where the error u is assumed to be distributed independently with mean The population regression model is: y = β 1 The remainder of the ANOVA table is described in more detail in Excel: (which equals R 2 given in the regression Statistics table). R 2 = 1 - Residual SS / Total SS (general Where yhat i is the value of y i predicted = Residual (or error) sum of squares + Regression (or explained) sum The ANOVA (analysis of variance) table splits the sum of It is not to be confused with the standard error of y itself (fromĭescriptive statistics) or with the standard errors of the regression It is sometimes called the standard error of the regression. The standard error here refers to the estimated standard deviation (when squared gives correlation squared = 0.8 = R 2 ).ĭiscussed later under multiple regression. The Regression Statistics Table gives the overall goodness-of-fit Number of observations used in the regression (n) This is the sample estimate of the standard deviation of the The regression output has three components:Īdjusted R 2 used if more than one x variable Regression of CARS on HH SIZE led to the following Excel output: This requires the Data Analysis Add-in: see Excel 2007: Access and Activating theĢ007: Two-Variable Regression using Data Analysis Add-in REGRESSION USING THE DATA ANALYSIS ADD-IN This handout is the place to go to for statistical inference for
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