In the multiple regression of the previous question, the F-statistic for the overall significance of the model has an

Question

BlackTom AI · Accepted Answer

To evaluate the F-statistic for the overall significance of a multiple regression model, we need to know the degrees of freedom used in the F distribution.
Option a: 'a. F4,40F_{4,40} distribution under the null' implies there are 4 regressors being tested (df1 = 4) and 40 degrees of freedom for error (df2 = 40). If the model actually has 3 predictors plus an intercept, df1 would be 3, not 4, so this option misstates the numerator df.
Option b: 'b. F_{3,41} distribution under the null' uses df1 = 3 but df2 = 41. The df2 value should reflect the total sample size minus the number of estimated parameters (including the intercept). If there are 3 predictors and an intercept, and with n observations such that df2 = n − 3 − 1 = n − 4, 41 would require n = 45. Unless the sample size is exactly 45, this is inconsistent with the common setup.
Option c: 'c. F_{4,44} distribution under the null' has df1 = 4 and df2 = 44. This would correspond to 4 estimated parameters plus an intercept (i.e., 4 predictors) and n such that df2 = n − 4 − 1 = 44, which would give n = 49. If the model only has 3 predictors, this is not correct.
Option d: 'd. F_{3,40} distribution under the null' uses df1 = 3 and df2 = 40. This matches the typical case where the model has 3 predictors (k = 3) and the error degrees of freedom equal n − k − 1. If the sample size is n = 44, then df2 = 44 − 3 − 1 = 40, which is a standard configuration for a model with three regressors and an intercept.
Option e: 'e. F_{3,44} distribution under the null' would imply df2 = 44, which would require n = 48 (since df2 = n − k − 1 = n − 3 − 1). If the sample size is not 48, this would be inconsistent with the model’s degrees of freedom.
In summary, each option assigns a pair of degrees of freedom to the F-distribution that must align with the number of predictors in the model (k) and the total sample size (n) via df1 = k and df2 = n − k − 1. The most conventional and internally consistent pairing for a model with 3 predictors and a typical sample size that yields df2 = 40 is df1 = 3 and df2 = 40, i.e., F_{3,40}.
Thus, option d aligns with the standard formula df1 = number of regressors and df2 = n − k − 1 under the null hypothesis that all regression coefficients (except the intercept) are zero, while the other options either misstate the numerator degrees of freedom (df1) or require a sample size that does not fit the assumed model structure. In particular, misalignments such as df1 = 4 or df2 values that imply incompatible n values lead to incorrect characterizations of the F-statistic under the null.

In the multiple regression of the previous question, the F-statistic for the overall significance of the model has an单项选择题

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