Confidence intervals - Statistics AP Study Notes

Think of a confidence interval like trying to hit a target with a dart, but instead of aiming for a tiny dot, you're aiming for a whole section of the dartboard. You're not saying, "I'll hit exactly the bullseye," but rather, "I'm pretty confident I'll hit somewhere within this big circle around the bullseye."

In statistics, that "bullseye" is the true answer for a whole big group (like all teenagers in the country), which we call the population parameter (the real percentage of all teenagers who love pizza). Since we can't usually measure everyone, we take a sample (a smaller group, like 100 random teenagers).

From our sample, we get a point estimate (our best single guess, like "70% of our sample loves pizza"). But we know our sample might not be perfectly representative. So, a confidence interval gives us a range (like 67% to 73%) where we're pretty sure the true population parameter lies. It's our way of saying, "We're X% confident the real answer is somewhere in here!"

• Confidence Level: This is how "confident" you want to be. Common levels are 90%, 95%, or 99%. A 95% confidence level means if you did this dart-throwing experiment 100 times, your "big circle" would successfully capture the bullseye 95 times out of 100. It's not the probability that the true parameter is in this specific interval, but about the method working over and over.
• Margin of Error: This is the "wiggle room" around your point estimate. It's how much you add and subtract to your best guess to get the upper and lower bounds of your interval. It's like how big your "big circle" on the dartboard is.

Let's say a big video game company wants to know what percentage of all gamers (the population) would buy their new game. They can't ask every single gamer in the world, so they pick 1,000 random gamers (their sample).

1. Survey Time! They ask these 1,000 gamers, and 600 of them say they would buy the new game.
2. Point Estimate: Their best single guess (their point estimate) is 600/1000 = 60%, or 0.60.
3. Building the Interval: Now, they know 60% is just from their sample. It's probably not exactly the percentage for all gamers. So, they calculate a confidence interval. Let's say they want to be 95% confident.
4. The Result: After doing some calculations (which we'll learn later!), they might find their 95% confidence interval is (0.57, 0.63). This means they are 95% confident that the true percentage of all gamers who would buy their new game is somewhere between 57% and 63%.

This helps them decide whether to launch the game, how much to spend on marketing, and how many copies to produce. If the interval is too low, they might rethink their strategy!