Any Statistics Guys Out There?

I apologize in advance for the rare non-gaming post and assure you it will not be a frequent occurence.  But I have a question for any statisticians out there.  Here goes.

Every week, I choose 10-15 professional football games to predict.  What I do is select who I think will cover the established gambling point spread.  So I don't pick winners or losers.  I try to pick who will cover the spread.  In theory, this should be a 50% proposition.  So if I was a trained monkey randomly picking which team will cover the spread, I should get about 50% of the games right.  In fact, Las Vegas sets the spreads so that approximately 50% of the money gets placed on each team.

Well, the season is over and my picking record is 119 wins and 83 losses over the course of 202 games.  That is, I correctly predicted which team would cover the point spread 59% of the time.  I was thrilled with the result and bragged to my brother about it.  I told him that most NFL analysts on TV don't pick point spread winners correctly 59% of the time.

But Dominic (my brother) rained on my parade by saying his gut instinct told him it was not a statistically meaningful sample since I only predicted 202 games.  I told him I thought that was enough games to draw some inferences about my handicapping ability!  I said "Dom, if I was to flip a coin 202 times I doubt it would come up on a particular side 59% of the time."

So does anyone know if 202 games is enough to draw a meaningful conclusion about my ability to pick games?  Or could it be pure luck?  I promise my feelings won't be hurt!

PS: I didn't actually bet a single penny.  I'm not really a gambler.  This was all to see how well I could do at picking point spread winners.