Can someone interpret me the line in the bold ? how it is different from p-value as p-value is also responsible for the same 5% error ? The text is from "An Introduction to Statistical Learning with Applications in R" by Trevor Hastie and Robert Tibshirani
Given each individual p-values for each variable, why do we need to look at the overall F-statistic?
After all, it seems likely that if any one of the p-values for the individual variables is very small, then at least one of the predictors is related to the response. However, this logic is ﬂawed, especially when the number of predictors p is large.
For instance, consider an example in which p = 100 and H0 : β1 = β2 = ...= βp = 0 is true, so no variable is truly associated with the response. In this situation, about 5% of the p-values associated with each variable (of the type shown in Table 3.4) will be below 0.05 by chance. In other words, we expect to see approximately ﬁve small p-values even in the absence of any true association between the predictors and the response. In fact, we are almost guaranteed that we will observe at least one p-value below 0.05 by chance!
Hence, if we use the individual t-statistics and associated pvalues in order to decide whether or not there is any association between the variables and the response, there is a very high chance that we will incorrectly conclude that there is a relationship.
However, the F-statistic does not suﬀer from this problem because it adjusts for the number of predictors. Hence, if H0 is true, there is only a 5% chance that the Fstatistic will result in a p-value below 0.05, regardless of the number of predictors or the number of observations