Wow, sounds like a lot of overprotective parents in here. 50 years ago it was not uncommon for kids that age (and younger) to actually play outside all day, roaming the neighborhood, with no adult supervision. Not even cell phones! On weekends, most kids were kicked out of the house in the morning and told not to come back until dinner. And you know what? Those kids were just fine.
I know, I know, times have changed. Well, guess what, kidnapping, violent crime, and theft are MUCH lower now then they were back then. The only thing that has gotten worse is the introduction of the 24-hr news cycle. I know you hear scary stories on the news about kidnapping, but only 0.014% of kids that are reported missing were actually kidnapped by a stranger. In the entire country, that's just 115 kids per year. Your kids are 5 times more likely to get struck by lightning.
Of the other 800,000 kids reported missing, 33% were kidnapped by a family member or friend, or 261,200 per year. The rest were runaways, kicked out of home, or lost on their own. If you think about it, since so many kidnappings were by family and friends, your kid is over 2000 times more likely to be kidnapped if they are accompanied than if they are alone. See
http://www.cnn.com/CNN/Programs/anderson.cooper.360/blog/2007/01/raw-data-kidnapping-statistics.html for the raw numbers.
DISCLAIMER: I'm not suggesting that all 7-year-olds can handle being in a theme park alone. Some of course can't, but on the other hand some can. Maturity, independence, responsibility, and self-reliance vary from child to child. However, that decision should be left up to the parents of the child, not posters on a message board.