Confidence Interval: Trash In Waterways

Hey guys! Today, we're diving into the fascinating world of statistics, specifically focusing on how to calculate a confidence interval for a population proportion. Imagine we're environmental scientists, and we've taken a random sample of waterways near large cities to check for trash. This is a common scenario in many fields, from public health to market research. Understanding how to construct a confidence interval allows us to make informed decisions and draw conclusions about the larger population based on our sample data. It's like having a superpower that lets us peek into the true state of things, even when we can't examine everything directly!

In this case, we tested 22 waterways and found that 13 of them contained trash. Now, the big question is: what can we say about the proportion of all waterways near large cities that contain trash? We can't test every single waterway, of course, but using statistical methods, we can estimate a range within which the true proportion likely lies. That range is what we call a confidence interval. In essence, we're building a net to catch the true population proportion, and the confidence level (in our case, 70%) tells us how sure we are that our net will actually capture it.

This process involves several key steps, from identifying our sample proportion to calculating the margin of error and finally constructing the interval. We'll be using the normal distribution as an approximation, which is valid when our sample size is large enough. This is because of the Central Limit Theorem, a cornerstone of statistical inference. This theorem essentially states that the distribution of sample means (or proportions) will approach a normal distribution as the sample size increases, regardless of the shape of the original population distribution. This is super handy because it allows us to use the well-understood properties of the normal distribution to make inferences about our population.

So, grab your calculators (or your favorite statistical software), and let's embark on this journey to uncover the secrets hidden within our data! We'll break down each step in detail, ensuring you not only understand the mechanics but also the underlying logic behind constructing a confidence interval. By the end of this discussion, you'll be equipped to tackle similar problems and confidently interpret the results. Think of it as leveling up your statistical skills – you'll be able to analyze data and make informed decisions like a pro!

The first step in constructing a confidence interval is to calculate the sample proportion, often denoted as p̂ (pronounced