So far, we have been focusing our attention on doing inference for one population mean. We usually have data from one sample way to find the sample mean and the sample standard deviation and use those values to be able to say something meaningful about the unknown population mean. However, the methods that we're learning are actually really not limited to just this one special case. We can actually use the same methods for doing inference for a variety of other estimators. We're going to be learning about these in the next two units in more detail, but this video is kind of a peek into the unified nature of hypothesis testing and confidence intervals. The methods that we've been learning for, doing hypothesis tests and constructing confidence intervals, can be easily adapted for any estimator that has a nearly normal sampling distribution. One example that we've been working with is a sample mean. Another example that's useful is the difference between two sample means. So, this type of an estimator would be useful for comparing two groups of population means, for example. Another estimator that might be of interest is the sample proportion. The sampling distribution of the sample proportion will also be nearly normal, as long as our sample size is high. And then we can apply the same techniques that we've been learning to do inference for proportion, or even also looking at difference between two proportions. So again, this gives us an avenue by which to con compare two groups to each other, two populations to each other. An important assumption about the point estimates is that they're unbiased. In other words, the sampling distribution of the estimate is centered at the true population parameter it estimates. That is, an unbiased estimate does not naturally over or underestimate the parameter but instead it provides a good estimate. We know that the sample mean is an es, example of an unbiased point estimate, because the central limit theorem tells us that the sampling distribution of sample means is going to be nearly normal centered at the true population mean. And the other estimates that we listed in the previous slides, are also good examples of unbiased estimators. So, if we have a point estimate that we know is unbiased and that has a nearly normal sampling distribution, then we already know how to construct a confidence interval around it. We always start with a point estimate when working with confidence intervals. And then we add and subtract the same amount to that point estimate. This is kind of the leeway we're giving ourselves when we're doing this estimation. And the value that we add and subtract we said, is the margin of error. The margin of error is comprised of two components. The critical value which is a z star, if we're working with a nearly normal sampling distribution, and the other one is the standard error. So, the one thing we're not going to get into in this video, is how to find the standard error for different types of point estimates. Because that's going to be the focus of what we're going to be doing in the next two units. So, once you have this general structure set up for the confidence interval, all you need to do is to swap out the formula for the standard error for a different estimate, but you keep everything else the same. In other words, with what you have learned so far you already know how to calculate confidence intervals for a variety of point estimates that happen to have nearly normal sampling distributions. What you're still looking forward to finding out though, is how to calculate the specific standard errors for those. So we are going to give a couple of examples working on constructing confidence intervals and doing hypothesis tests for these different different point estimates. But, we are just going to give away the standard error to you in this particular video and then we are going to get into more detail in the following units. Let's take a look at a practice problem. A 2010 Pew Research foundation poll indicates that among 1,099 college graduates, 33% watch the Daily Show. An American late-night TV Show. The standard error of this estimate is 0.014. We are asked to estimate the 95% confidence interval for the proportion of college graduates who watch The Daily Show. Let's start by parsing through some of the information we are given. The 33% who watch the daily show among the, these observed college graduates is going to be our p hat 0.033. P hat stands for sample proportion, just like x bar stands for sample mean. And we are also told that the standard error of this estimate is 0.014, so let's take a note of that as well. By now, we know the generic formula for a confidence interval for any estimator. It's always a point estimate, plus or minus a margin of error. In this case, our point estimate is a p hat, and then we have plus or minus a critical value, z star, times our standard error, that make up the margin of error. The p hat is 0.33 plus or minus 1.96 for the critical value, times the standard error that we're given in the problem. Gives us a margin of error of 0.027 or 2.7%. Adding and subtracting that to our point estimate, we get a confidence interval that says that we are 95% confident that between 30.3% and 35.7% of college graduates watch the Daily Show. Just like with confidence intervals, we can apply the same framework for hypothesis testing to different estimators, as well. And again, as long as the estimator is unbiased and has a nearly normal sampling distribution. So if that's the case, we can use the z statistic as our test statistic, that we always calculate as a point estimate minus the null value, kind of like the observed minus the mean, divided by some standard error. And we're not, again, once again, going to get into the, calculating the standard error for these different point estimators, but that's something we're going to focus on in the following units. Now let's take a look at a practice problem doing a hypothesis test on an estimator different than the sample mean. The third national health and nutrition examination survey NHANES, collected body fat percentage and gender data from over 13,000 subjects in ages between 20 to 80. The average body fat percentage for the 6,580 men in the sample was 23.9%. And this value was 35% for the, for the 7,021 women. The standard error for the difference between the average male and female body fat percentages was 0.114. Do these data provide convincing evidence that men and women have different average body fat percentages? You may assume that the distribution of the point estimate is nearly normal. So now that we know that the distribution of the point estimate is going to be nearly normal, we know we can use the same framework we've learned before, to do this hypothesis test. And let's follow the same steps for doing so then. First we want to set our hypotheses. The null hypothesis is going to be that there is no difference between these two populations, so the null hypothesis is always a status quo. So that means that the average men and average women body fat percentage is equal to each other. And the alternative is going to speak to our research question. Do these data provide convincing evidence that men and women have different average body fat percentages? So the alternative is going to be two-sided. Our point estimate is simply the x bar version of what we have on our, in our hypothesis. So that's going to be the observed average body fat percentage for men, minus the observed average body fat percentage for women which comes out to be negative 11.1. Lastly, we're going to need to check conditions, but we're told that we can assume that the distribution of the point estimate is nearly normal. So we're safe on that account, and since this is a nationwide survey, I think we can be reasonably certain that they have used random sampling and such that the observations in the sample are independent of each other with respect to their body fat percentages. And next, what we want to do is to be able to draw our curve. But before we can do that, we need to figure out what the sampling distribution of this estimator looks like. We know the shape. It is nearly normal, but what is the center going to be? The center is usually the null value. However, in our null hypothesis, there is currently no value. So we can rewrite our null hypothesis as the difference between the two population means being equal to zero. Because after all, if the two quantities are equal to each other, then their difference is going to be zero. Which tells us that the sampling distribution is nearly normal and centered at zero. And our p value is going to be any region that's beyond the out, observed difference between the two means. So that could be less than negative 11.1, or greater than positive 11.1. I think its looking like this is going to be a pretty tiny p value, because the shaded regions are so small, but for completeness once again we can actually calculate our z score. So the z score is calculated as the point estimate minus the null value divided by the standard error and that is one huge z score of 97.36. With such a huge z score, the p value is bound to be really, really tiny which is going to result in us rejecting the null hypothesis. And in context, what we would then determine, is that these data provide convincing evidence that the average body fat percentages of men and women are indeed different from each other.