Sports and HDTV on the HC3700
Fire up Bright Cinema mode for your sports viewing (and for most HDTV if you have ambient light present, and the HC3700 should not disappoint. There's plenty of lumens to go around, so the HC3700 can tackle more than a little ambient light, although if you have a ton of ambient light you likely should be looking at we dub "bright room" projectors.
Colors are rich looking and well saturated (like all 3LCD projectors, they have equal numbers of color and white lumens. (We don't measure, we take Epson's word, because we have measured Epson projectors in the past). One expects equal color and white lumens from basically all 3 panel / chip projectors, be they 3LCD or LCoS. It's the single chip DLP projectors that normally come up way short on color lumens. I used the HC3700 for some football viewing during this year's NFL playoffs and have no complaints. I turned on CFI aka "smooth motion" to Epson's low setting which I found to do a bit of smoothing, without noticeable artifacts.
For the non-sports viewing, I normally turn off CFI, as I'm not that big a fan of creative frame interpolation. I consider it a "nice extra" if you like it, but sometimes I find it "over the top" even on normal HDTV viewing like sitcoms, Blacklist, Westward, etc. In all cases I find the High setting to be too much, as one can spot artifacts around fast moving objects in the image.
Epson's Bright Cinema mode is just fine for just about all my HDTV viewing, although when there are a lot of well produced skin tones (such as the Victoria Secret shows), I will use my calibrated Cinema mode. The non-critical viewer may prefer the slight extra punch of Bright Cinema over Cinema, thanks to only a small penalty in the naturalness of those skin tones.
All considered, the Home Cinema 3700 has plenty of horsepower to tackle your HDTV and sports with modest to a moderate amount of ambient light, without major sacrifice. Count the 3700 as an excellent choice in the price range for "TV" viewing.
Bottom Line on Picture Quality
Epson's Home Cinema 3700 is an especially well balanced projector in it's price range about half-way between $1000 and $2000. Overall, color is impressively good right out of the box in three of the four modes. The fourth is Dynamic - the brightest mode - and we never have high expectations for any projectors's brightest mode, although this Epson's color balance in Dynamic is a lot better than many other projectors.
I find this Epson to be great on sports viewing and almost equally good on general HDTV. Movies only suffer from the less than stellar black level performance, but that is a trait of virtually all under $2000 projectors even if a few do a little better than the HC3700. I would have to jump up $500 to Sony's HW45ES to claim a better overall projector in terms of picture quality. There are a couple of very good DLP projectors as well (the ones with the best black levels in the price range), but this Epson typically has the advantage on those in terms of handling sports and HDTV with any ambient light present.