Classic linear model - model selection I have a classic linear model, with 5 possible regressors. They are uncorrelated with one another, and have quite low correlation with the response. I have arrived at a model where 3 of the regressors have significant coefficients for their t statistic (p<0.05). Adding either or both of the remaining 2 variables gives p values >0.05 for the t statistic, for the added variables. This leads me to believe the 3 variable model is "best".
However, using the anova(a,b) command in R where a is the 3 variable model and b is the full model, the p value for the F statistic is < 0.05, which tells me to prefer the full model over the 3 variable model. How can I reconcile these apparent contradictions ?
Thanks
PS
Edit: Some further background. This is homework so I won't post details, but we are not given details of what the regressors represent - they are just numbered 1 to 5. We are asked to "derive an appropriate model, giving justification". 
 A: One answer would be "this cannot be done without subject matter knowledge". Unfortunately, that would likely get you an F on your assignment. Unless I was your professor. Then it would get an A.
But, given your statement that $R^2$ is 0.03 and there are low correlations among all variables, I'm puzzled that any model is significant at all. What is N? I'm guessing it's very large. 
Then there's 

all the 5 predictors are generated by independent simulations from a
  normal distribution.

Well, if you KNOW this (that is, your instructor told you) and if  by "independent" you mean "not related to the DV" then you know that the best model is one with no predictors, and your intuition is correct.
A: You might try doing cross validation. Choose a subset of your sample, find the "best" model for that subset using F or t tests, then apply it to the full data set (full cross validation can get more complicated than this, but this would be a good start). This helps to alleviate some of the stepwise testing problems. 
See A Note on Screening Regression Equations by David Freedman for a cute little simulation of this idea.
A: I really like the method used in the caret package: recursive feature elimination.   You can read more about it in the vignette, but here's the basic process:

The basic idea is to use a criteria (such as t statistics) to eliminate unimportant variables and see how that improves the predictive accuracy of the model.  You wrap the entire thing in a resampling loop, such as cross-validation.  Here is an example, using a linear model to rank variables in a manner similar to what you've described:
#Setup
set.seed(1)
p1 <- rnorm(50)
p2 <- rnorm(50)
p3 <- rnorm(50)
p4 <- rnorm(50)
p5 <- rnorm(50)
y <- 4*rnorm(50)+p1+p2-p5

#Select Variables
require(caret)
X <- data.frame(p1,p2,p3,p4,p5)
RFE <- rfe(X,y, sizes = seq(1,5), rfeControl = rfeControl(
                    functions = lmFuncs,
                    method = "repeatedcv")
                )
RFE
plot(RFE)

#Fit linear model and compare
fmla <- as.formula(paste("y ~ ", paste(RFE$optVariables, collapse= "+")))
fullmodel <- lm(y~p1+p2+p3+p4+p5,data.frame(y,p1,p2,p3,p4,p5))
reducedmodel <- lm(fmla,data.frame(y,p1,p2,p3,p4,p5))
summary(fullmodel)
summary(reducedmodel)

In this example, the algorythm detects that there are 3 "important" variables, but it only gets 2 of them.
A: The problem began when you sought a reduced model and used the data rather than subject matter knowledge to pick the predictors.  Stepwise variable selection without simultaneous shinkage to penalize for variable selection, though often used, is an invalid approach.  Much has been written about this.  There is no reason to trust that the 3-variable model is "best" and there is no reason not to use the original list of pre-specified predictors.  P-values computed after using P-values to select variables is not valid.  This has been called "double dipping" in the functional imaging literature.
Here is an analogy.  Suppose one is interested in comparing 6 treatments, but uses pairwise t-tests to pick which treatments are "different", resulting in a reduced set of 4 treatments.  The analyst then tests for an overall difference with 3 degrees of freedom.  This F test will have inflated type I error.  The original F test with 5 d.f. is quite valid.
See http://www.stata.com/support/faqs/stat/stepwise.html and stepwise-regression for more information.
