 Hi, I'm Mike Marin. Today, we'll discuss the concept of a confidence interval for a population mean. When we do research, we often collect a sample, and then we use that data to calculate an estimate, say, the mean of a variable. But what if our guess for the population mean relied solely on the sample mean we got today? Wouldn't relying on this single value be problematic? Wouldn't other samples we took tomorrow be different? The sample mean is our best guess at the population mean, but yes, we know that this estimate will differ a bit from the true value, due to what we call sampling variability. To account for differences that would occur between samples taken today, yesterday, or tomorrow, we can attach a margin of error to our estimate and be reasonably confident that we will capture the population mean. This is the idea behind a confidence interval. Suppose you wanted to take a sample to estimate the mean length of fish in a lake. And bear with me for a moment, but suppose somehow you could know the true population mean length of fish in this lake. While our sample mean should be reasonably close to that true mean of 40 cm, we know that due to sampling variability, it won't be exactly equal to 40. Our estimate is just one of many possible sample means we could have ended up with on any particular day. And the sampling distribution describes this theoretical set of all possible estimates. Understanding the concept of the sampling distribution of the mean helps us attach a margin of error to our estimate and create a confidence interval. Man, it's getting late. Rules of thumb are helpful things. And you should remember that we have one rule of thumb that tells us that under certain conditions, about 95% of all the sample means we could end up with will be less than two standard errors away from the true mean. And if a sample mean is usually less than two standard errors away from the true mean, then the true mean is usually less than two standard errors away from the sample mean. In other words, if we reach out about two standard errors from a sample mean, we will usually capture the population mean in this interval. And using the same rule of thumb, we know that about 5% of sample means will be more than two standard errors away from the true mean. And in this instance, reaching out two standard errors from the sample mean will not capture the true mean. Of course, we will never know how far our estimate truly is from the true mean. We're only humble statisticians, and we must accept this. We must accept our limitations. Now, let's get the heck out of here and take a look at an example. Suppose you are interested in estimating the mean length of fish in this lake, and you only have this lousy, lazy paper cut out of a fisherman to help you. Let's try something different to take a random sample of 25 fish from this lake. For these 25 fish, you find they have a sample mean length of 41 cm and a sample standard deviation of 12 cm. Now, your best guess at the true mean length of all the fish in this lake is 41 cm, but this, my friends, is where we can bring in our margin of error. We can start from our sample mean and reach out two standard errors, which is our margin of error here, on each side of the estimate to create a 95% confidence interval. We would then say that we're 95% confident that the true mean length of fish in this lake is between 36.2 to 45.8 cm, and this, my friends, is our confidence interval. But remember, 5% of such intervals will not contain the true mean. We will never know if our confidence interval actually captures the true mean or not, and we're okay with that. Also remember, the standard error and hence the width of the confidence interval will depend on the sample size. Holding all else constant, as the sample size gets larger, the standard error will become smaller and the confidence interval will become narrower or more precise. Pretty interesting stuff, eh? In addition to estimating the mean of a population, we're often interested in estimates that summarize the relationship between two or more variables, such as differences in means or correlations and things like these. We can construct confidence intervals for these as well, and they are all based on the same concepts and principles discussed here, and all have the same general interpretation. In a separate video, we'll discuss how the sampling distribution can be used to test the hypothesis about a population mean. Don't forget to check out the statistics visualizations that accompany this video. You can find the link in the description. Thanks for watching! I didn't even get the money. I moved up to the road. I've been wandering to where they'll throw and made my way to the car.