Why does the public accept government produced projections or statistics when they are generally inaccurate?

Projections are informed guesses and are expected to be revised, and are nearly always questioned as soon as they are publicized. Statistics are a little different, and are rarely adjusted by very much. If the government finds that unemployment was 8.1% in a month and then figure out that it was really 8.0% or 8.2% and corrects it, I would expect little fanfare--the story is the same, unemployment a bit over 8% and not moving up or down by enough to really matter. Sure, if you want to be a democrat claiming a victory every time those figures move down slightly or a republican saying the dems are moving us backward every time they inch up you can, but take a look at reality. That figure hasn't changed much at all in quite a while--people would be more right to say it barely moved and talk about why that is a problem, but instead they want some irrelevant statistics--got worse this many months, improved this many months--basically anything that makes a story.

I think your best source for information about the government is still the government. Go straight to the congressional budget office-- don't trust what Obama or Romney say (unless you listen extremely carefully to what they say, as they cherry pick figures to make their side look good). Unfortunately, you can't really trust what the news media say either--they are trying to shock and entertain you--consider their articles a place to start looking for the truth, not a place to find it. The thing that CBO does that campaigns and news organizations do not--they look at the same types of numbers, calculated the same way, all the time for years and years. Sometimes there will be an adjustment if they get new information, but so what--you'll always get the figures they think are accurate at the time. They leave the raw numbers on the internet for all to see and tell you how they came up with them. After the CBO publishes numbers, then the press and politicians pick their favorite parts of the numbers and try to figure out how to turn them into something shocking enough to print or stupid enough to throw up in a campaign.

In short, if you read the news, that version of the truth will need frequent and major revision. If you read past the news (see what was said and look up the sources and get numbers to confirm it) and look at what is really going on, half the 'problems' with the democrats and the republicans are someone making mountains out of molehills. The really shocking thing is that when you do this, you often find that whatever the article was ranting about is no big deal at all--but you see other things that are very important and totally ignored while you check up on the article.

In order to answer intelligently, you'd have to provide a specific example. Some government statistics are more accurate than others. It depends, as you say, how the numbers are determined.

Projections, by their very nature, are going to be of limited accuracy because no one has a crystal ball, and many assumptions have to be made in order to forecast the future.

I wouldn't trust most of what politicians say (of any party). Non-partisan government agencies are usually reasonable sources of information, and are often relatively transparent in terms of their methodology and assumptions.