Short answer is yes  

I started out a month ago playing violin when I received my Stagg violin outfit which of course is simply rubbish (worse than your Cecilio). 

The bow that came with it has a wooden stick, which actually looks decent at first glance since it's not crooked and has the correct arch, but the hair on it is nylon (synthetic)

Just for that reason alone I wanted to order a new bow that has horse hair, but the company I ordered it from had only a single piece at an affordable price range (and even that was in one of their remote shops), which delayed my order for a month.. after which they 'lost track of it'.. so long story short.. we decided to downgrade to a cheaper model that they actually had.

So I received that and was disappointed to see that this one also has synthetic hair on it.. After rosining it up and playing with it though I instantly realized that it's a MAJOR upgrade compared to what I had..

The balance was way better.. the old one felt like swinging a baseball bat compared to the new bow (and I'm not kidding), the tip was wider (actually the tip on the old bow was narrower than standard) allowing for more space between the stick and the hair without over-tightening 

I don't know what kind of bow do Cecilio violins come with, but if there's THIS much difference between 'nylon-bow' and 'nylon-bow' then you'll probably want to give it a try.

Now for the question of which bow... Upon roaming the internet for information on bows in the past few weeks the conclusion I got from all of it is this: Generally at a lower-to-mid price range, carbon fiber pretty much blows everything out of the water, since it can be engineered to perfect specifications with optimal weight balance etc... (there's no 'reason' to make them bad, since they're built from the ground up unlike wooden bows where even the worst quality wood is probably used by some off-the-radar factories during their night-shift ).

You'll also find fiberglass bows at a lower price range, which are generally regarded as garbage and everyone seems to want to re-purpose them as tomato-sticks for the garden.. so you might want to avoid those.

You'll also find barzil-wood bows. Interesting info: While Pernambuco IS Brazil-wood (a type of wood that grows in Brazil), Brazil-wood isn't Pernambuco Meaning that once it started to get rare and endangered (a few centuries ago?), manufacturers started using other types of wood (still species that grow in Brazil) as replacements and it pretty much dropped the barre on quality. This means that now we have brazil-wood bows which CAN be decent, however most are student-quality and usually inferior to carbon-fiber and we have Pernambuco which is usually specified and isn't 'mixed' in with all the other brazil-wood (especially since the price is like 10 times) and that's probably the only thing that could surpass carbon fiber (has to do with how it resonates, adding to the sound - which can probably only be heard by people who really want to hear it lol).

Of course these aren't rules set in stone.. every person is different and the balance of a bow that I hate for example.. might be just what YOU are looking for haha.. that's why it's probably better to hold it in your hand before buying or at least have some money-back guarantee options if buying online. 

But anyway... I think you might want to try Fiddlerman's Carbon Fiber bow.. I know I would love to try that if I had a chance Everyone seems to love it and the price doesn't look that wild to me... Otherwise if you plan on getting a new violin as well, you might want to get a whole outfit from him, since they usually include that CF bow with their outfits at an even better price or you could get just the bow and then a violin without an outfit later.. up to you  

Hope this helps

After posting that I actually went ahead and ordered one for myself lol

Okay.. so my bow just arrived..

I am slightly confused because they had me pay import taxes for the shipping fee as well.. which is strange... since I am not importing 'shipping'... Even though I sent them the paypal screenshot which lists them separately, they added it to the price of the bow and listed the whole amount I paid Fiddlershop as the product price on the customs document. Maybe that's how it's supposed to be done.. who knows  

Other than that, the bow itself survived the trip and is really nice. Doesn't shake as much as the old one I had so it produces a more stable sound. Overall I like it.. it does have some aesthetic flaws at the grip, but that's okay I guess  

I do recommend it, since it's better than most bows that come with beginner outfits. 

On this bow I used the pirastro goldflex, since I wanted to try it out separately without mixing I kinda mixed it with a thomastik medium rosin on the other bows