There are some potential problems with a multiple regression analysis: 1. The critical value from the t table will have n − k − 1 degrees of freedom. Confidence intervals and hypothesis tests for an individual regression coefficient will be based on its standard error, S b 1, S b 2 ,…, or S b k. If the regression is significant, you may proceed with statistical inference using t tests for individual regression coefficients. If your regression is not significant, you are not permitted to go further. Inference begins with the F test, an overall test to see if the X variables explain a significant amount of the variation in Y. The coefficient of determination, R 2, indicates the percentage of the variation in Y that is “explained by” or “attributed to” the X variables. The standard error of estimate, S e, indicates the approximate size of the prediction errors. There are two ways of summarizing how good the regression analysis is. The prediction errors or residuals are given by Y − (Predicted Y). These coefficients ( a, b 1, b 2, … , b k) are traditionally computed using the method of least squares, which minimizes the sum of the squared prediction errors. Taken together, these regression coefficients give you the prediction equation or regression equation, Predicted Y = a + b 1 X 1 + b 2 X 2 + … + b kX k, which may be used for prediction or control. The regression coefficient b j, for the jth X variable, specifies the effect of X j on Y after adjusting for the other X variables b j indicates how much larger you expect Y to be for a case that is identical to another except for being one unit larger in X j. The intercept or constant term, a, gives the predicted (or “fitted”) value for Y when all X variables are 0. The goals of multiple regression are (1) to describe and understand the relationship, (2) to forecast (predict) a new observation, and (3) to adjust and control a process. Siegel, in Practical Business Statistics�(Sixth Edition), 2012 SummaryĮxplaining or predicting a single Y variable from two or more X variables is called multiple regression. The supplier differences are highly significant ( p < 0.01).Īndrew F. Since the F statistic is larger than the critical F value, the result is highly significant, a stronger result than before: To test supplier quality at the 1% level, the F statistic (5.897) is compared to the critical F value (somewhere between 5.390 and 4.977 from the F table or, more exactly, 5.013). There are significant differences among your suppliers in terms of average quality level ( p < 0.05). Since the F statistic is larger, the result is significant: To test the supplier quality example at the 5% level, the F statistic (5.897) may be compared to the critical F value (somewhere between 3.316 and 3.150 from the F table or, more exactly, 3.165). (All of the above statements are equivalent to one another.) The observed differences among the sample averages could not reasonably be due to random chance alone. The sample averages are significantly different from each other. If the F statistic is larger than the critical F value: The result is not statistically significant. The observed differences among the sample averages could reasonably be due to random chance alone. The sample averages are not significantly different from each other. If the F statistic is smaller than the critical F value:Īccept the null hypothesis, H 0, as a reasonable possibility.ĭo not accept the research hypothesis, H 1.
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