I just bought a house and had it inspected before buying. The inspector was very thorough: down to the bolts on the door locks locks. He went into every nook and cranny. He took pictures all over the crawl space his inspection covered 78 pages in detail but oddly enough, he made no mention of a huge sump pump in the corner on the crawl space. It's right under my bedroom and every night I wondered what the neighbor was doing with a constant humming every 5 minutes. Then one day standing out side on a dry day, about 3 or 4 gallons of water comes gooshing out the drain pipe. I got curious and crawled under the house. It's very clean, white clean crawl space and found in the corner a Triplesafe sump pump system with Battery back up and now I'm very concerned, I don't know what recourse I have. The sump pump turns on about every 5 minutes for about a minute. It is dry under the house but can't this pump burn up or something? How deep is he well that it is connected to? How in the world could the inspector miss this?
Make sure that your home owners insurance covers water damage from a sump pump failure if the pump fails and damages an area of your house. A couple years ago, mine failed during a long wet spell and I called my insurance company and was told that even though I was covered from every other water disaster, a sump failure was not covered. (This disaster for me fell under the heading, "crap, that is how my insurance company classifies a loss as mine alone"). My finished basement flooding cost me a bundle and nothing was covered.
You typically can only get back what you paid the home inspector for his service. In other words, this is your problem now, you can forget about the inspector. The sump pump isn't your issue here. The problem is the water. Stop the water and you stop the noise and wear and tear on the pump.
One thing you should make sure of first is that there is a check valve on the outlet pipe. Since noise is an issue, you'll want a 'quiet' one. If you don't have one of these, a lot of the water can drain back from the pipe into the sump well so you'll end up pumping the same water more than once. If you don't have one, adding one can reduce the frequency of running it.
Aside from that, you need to find sources of water. Check your usage for one. Is it especially high? If you can see the meter, is it spinning even when no water is being used? What's the drainage like from the roof? Are neighboring yards draining to yours?
If this pump is pulling from the water table, you might be out of luck with resolving the constant running. Putting in soundproof insulation under your bedroom may be the only option.