Hello, welcome back, welcome to module 4. In this module we're going to be talking about confidence intervals. Now in most of this course so far, we've talked about estimators of perimeters a generic theta or a function tau theta. We've talked about alpha and beta for a gamma distribution mu and sigma squared for a normal distribution, lambda for an exponential lambda for a Poisson, P for the geometric. We've talked about how to come up with these estimators and how to measure error. We've talked about variance, which is a measure of how far away the estimator is from its mean and if it's unbiased for the thing you're trying to estimate, this is a measure of how far away you are from the thing you actually want and if it's not an unbiased estimator, then we stopped talking about variants and started talking about Mean Squared Error. But if we have an unbiased estimator, they are one and the same. If you are the statistician working for a scientist who has asked you to look at the data and estimate some things, should you come back with a point estimate of value and then tell them the variance. A variance on its own is really hard to interpret, it's easier to interpret relative to another variance, but I think instead of giving that scientist the estimate and the variance, you should give them a confidence interval, which is like a range of plausible values where you think the true parameter is. What is a confidence interval? I'm going to start off with a specific statement, suppose I say that a 95 percent confidence interval for the mean mu is negative 2.14 to 3.07. What does this mean? I'll tell you what it does not mean. First of all, it has nothing to do with your feelings, you are not 95 percent confident that the true mean is between negative 2.14 and 3.07. What does that even mean? Another thing, it does not mean that the true mean is between those two numbers 95 percent of the time because those two numbers are fixed and the true mean is out there, but it's fixed and it's either in there or it's not. There's no probability, where does the probability come in? We've already talked about this, it comes in from random sampling. The fact that if you select a random sample and compute your estimator and if you got to do it again or if someone else did it, they're going to get a different estimator based on the randomness of the sample, you collect your sample, you estimate your parameter, you use the techniques of this video to come up with a confidence interval, whatever that means, I still haven't defined it and if you did it again or if different people did it, they would get different samples and different results, there we get different intervals. Here is a number line and here is a mu that I'm estimating or it could be more generically a theta. I'm going to take a random sample, I'm going to use the techniques of this video and we're going to compute a confidence interval and it comes out to be this. Now, as I was saying, multiple samples give multiple confidence intervals, if I take another sample and use the exact same procedure, I may get this, and if I took another sample and did the exact same procedure, I may get this, those green parentheses for the last one, they don't even include the parameter mu. But what I can say is that if I keep doing this in the long run, 95 percent of them are going to correctly capture mu, there's a mu and interval, 5 percent of them, if you do what we're doing in this video, will miss that true mean mu, but 95 percent of them will capture it. We are going to build our confidence intervals from the ground up, I am not going to tell you how to do it and then show you an example, we're going to start right out with an example. I also want to emphasize that I think this is how you should be thinking about most statistical computations, not a formula in a box at the end, but knowing what you're doing so that you can figure it out when something is slightly different in your particular situation. Suppose I have a random sample of size n from the normal distribution with mean mu and variance sigma squared, I'm going to suppose that the mean is unknown, but the variance is known. Which sounds a little crazy. Why would you know the variance if you didn't know the mean? It is possible to make a case for this to say that I am selling cereal, and every year I changed the design on the box, and that drive sales. Over the last decade I've seen that the sales go up, and they go down, but the variance seems to stay the same. I want to take a sample from this year, and figure out the true mean sales of my cereal this year, that's a little contrived. What I'm going to tell you is that the sigma squared is assumed to be known as a building block, we'll eventually get rid of that assumption. Estimating mu, I want an estimator, x-bar is an estimator. I want to know its distribution, it is normally distributed with mean mu, and variance sigma squared over n. I'm not using the central limit theorem here, I am using the fact that we're starting with the normal distribution, and linear combinations of normals are normal. Then I know the mean, and variance for x bar no matter what the distribution. I'm going to take this information, and I'm going to standardize x-bar to turn it into a standard normal. This is something we can call z, which is our preferred letter for a standard normal, and if you look at the standard normal PDF, I can find for you two numbers, and they're not unique. But I can find two numbers that capture z with probability 0.95. There is two numbers on this PDF that has area 0.95 in the middle. Now this doesn't have to be in the middle, I could capture 95 percent of the area somewhere else. I'll talk about that eventually, but how are we going to do this? The CDF, or the standard normal distribution, we've been calling it capital fee, and we don't know what enclosed form but I've talked about the idea that people have integrated it numerically, and put this stuff in tables, and that you can also these days you software to integrate this distribution. We have talked about finding the probability that z is less than, or equal to a particular number say 1.14, and how you would do that in R, you would type p-norm 1.14 to find the area to the left of 1.14. If I have the area first, and want to find a number, that's sort of an inverse problem, and we've talked about finding all the area to the left of a number. The first thing I want to do with this, is I want to translate this in the middle to the left problem. If I have area 0.95 in the middle, then I have area 0.05 divided by two in each tail, that is 0.025, and if I add on the lower tail here, then I wanted to find a number that has area 0.975 to the left. My colleague's question mark, and if you go back to the other slide here, the numbers I'm looking for it looks nice and symmetric, are question mark, and negative question mark. There is a command in R, it's the inverse of the p-norm command, and it's called q-norm. If you type q-norm 0.975, then it will return the number that cuts off at area to the left, and when I did this in R I got 1.96. We know that we can capture a standard normal between these two numbers, negative 1.96, and positive 1.96 with probability 0.95, and we know that our sample mean, when standardized, when you subtract the mean, and divide by the standard deviation acts just like z. These two numbers will work for that as well. The last step into coming up with a confidence interval is to solve this for mu in the middle. I'm going to move things around until I get mu isolated alone in the middle, and you see I've got two end points. This is weird from a probability perspective, if you haven't had a lot of probability, usually we have the random variable in the middle, and we talk about the probability that x is between one, and three. Here, the randomness is in the endpoints, the mu in the middle is constant, and fixed. These two end points form the 95 percent confidence interval. Here is our first confidence interval, this is 95 percent confidence interval for the mean mu of a normal distribution and that 95 percent, really is what gave us those specific numbers, 1.96, and negative 1.96. Let's talk about this a little more generally but first, a notation you might want to use. A more compact notation is to say x-bar plus, or minus 1.96 times sigma over the square root of n. Sigma squared was assumed known, sigma the square root is known, the sample size is always assumed known in this course. There are other courses that I hope you're going to go on to take in statistics where you're going to talk about choosing a sample size to get what you'd like to see. But let's talk about that 1.96. This is called a critical value. We use the words "critical value" to denote any number that cuts off a certain area under the curve of a PDF. In general, for a standard normal distribution, this is really common notation, but not completely universal, but really close. We are going to use a lowercase z_Alpha, this is not a random variable to denote the number that cuts off the area Alpha to the right for the standard normal curve. Y to the right and Alpha left, it has to do with historically where confidence intervals fell. They came up after hypothesis testing, which is something in the next course, if you're going to take that, and it's more historical reasons than anything else. In our case, with the 95 percent confidence intervals, we found the critical values negative 1.96 and positive 1.96. But in general, we're going to put area 1 minus Alpha in the middle. I know you probably want to put Alpha there. But again, for historical reasons, we want area 1 minus Alpha in the middle. Also, Alpha in statistics is usually thought of as something small. It's not quite an epsilon, but it's a small thing. The Alpha for a confidence interval is going to represent the area out in the tails. For a 100 times 1 minus Alpha percent confidence interval, I want to put area 1 minus Alpha in the middle. For example, in the 95 percent case, my Alpha was 0.05 and I put area 1 minus 0.05 or 0.95 in the middle. If I have area 1 minus Alpha in the middle, then I have a total area of Alpha in the two tails. If I cut them off symmetrically, then I've got Alpha/2 in both tails. This number is called, by our new definition, this is the critical value that we denote, lowercase c_Alpha/2. Now, by symmetry, the other one is negative lowercase c_Alpha/2. But we can also write it 1 minus Alpha/2. Because in our new notation, Z_1 minus Alpha/2 is a number that cuts off area 1 minus Alpha/2 to the right for the standard normal distribution, which means Alpha/2 to the left. We are not going to need this because of symmetry, but eventually, we're going to do confidence intervals for distributions that are not symmetric. In summary, if I have a random sample of size n from the normal distribution where mu is unknown, but sigma squared is known. A 100 times 1 minus Alpha percent confidence interval for mu is given by x bar plus or minus a z critical value with the subscript Alpha/2 times sigma over the square root of n. You will note that, everything we have done in this video is based on the fact that we started with a normal distribution. What did we do with that? We wanted a confidence interval for a parameter mu. We found an estimator x bar that we knew something about. We know the distribution of x bar is normal. Then we standardized it to something that didn't even involve the parameters. The variable Z had a standard normal distribution, mean 0, variance 1, nothing unknown there. Then we found critical values for that distribution or that random variable and then we translated it back to our x bar. But everything was really based on the fact that, x bar has a normal distribution. That was true because it was a linear combination of normals. But what if we start with something not normal or even unknown? By the central limit theorem, for a more general or even unknown distribution, if our sample size is large, which we usually take to be greater than 30, then x bar has a roughly normal distribution. If you have a sample of size n, and you know the variance, and you don't know the distribution, and you want a confidence interval for mu as long as your sample is large, you can use the central limit theorem to say that, x bar is approximately normal. Then you can go through other steps we just did, identical to what we did before to say that, an approximate large sample, 100 times 1 minus Alpha percent confidence interval for the mean is given by x bar plus or minus z_Alpha/2 times sigma over the square root of n, and we didn't even need normality. Let's look at an example. We have a supermarket chain that is trying to figure out whether or not they should add more organic offerings to their produce section. So they hire an external marketing firm to collect some data. Then they send it to us as a statisticians to make some conclusions. Not very many in this video because we only know one thing at this point. But we're going to make more conclusions as the rest of this course plays out. Suppose we took a random sample, or the marketing firm took a random sample of 200 customers from this particular region. They observed that the average amount spent on organic produce per person per month is $36. Suppose based on previous studies from previous years, we believe that the distribution of these amounts are normal. We know the variance, which is $5 per person. There's a variance of $5 and the spread of the amount per person per month spend on organic produce. I would like to return a 90 percent confidence interval for the true mean mu. The true average amount per month per person that anyone from that store is going to spend on organic produce in a month. I have this data, we got N equals 200 x-bar. Notice I've made it lowercase. This is not random but fixed at 36. It is observed and measured and not a random variable. We've got sigma squared somehow known to be five. So we're ready. We've got a formula here. We've got an X-bar plus or minus thing. The only thing we're going to need to do is find the appropriate critical value. Because things are normal here I'm going to look at the standard normal distribution. Really, if I was doing this from scratch, I would say, well, x-bar has a normal distribution and if I standardize it now I have a standard normal and I can find cutoffs for that and blah, blah, blah, blah, blah. For a standard normal distribution like this, this is the PDF for the normal 01, I would like to find two values that give us area 0.9 in the middle. This means that we have area 0.95 to the left of our question mark here. Then the other cutoff is the negative question mark. You can find this cutoff in R by typing qnorm(0.95). You'll get approximately 1.645. Our formula is X-bar plus or minus the critical value times the standard deviation over the square root of n. We've got an alpha of 0.1 because our confidence interval was 90 percent, one minus alpha 0.9. We know our z critical value and we have stuff to plug in. We do, and we end up with 35.74 for up to 36.26. Again, this does not mean that the true mean is in there 90 percent of the time, or that we feel 90 percent confident that it's in there. It's just an interval that 90 percent of the time, if you did our techniques, you would be right in including the true mean. In this video, I've talked about computing or capturing an area, a specified area in the middle between two numbers. I've taken those numbers to look symmetric about zero. There's no real reason to do that other than it's going to give you a shortest possible confidence interval. Try it out. If this end point moves over here, then this one has to move over here. Because there's less area over here, this one is going to have to move further over than this one move. That's going to increase the length of the interval. It would be a valid confidence interval in the end. But why give someone a bigger interval than you actually need? You're trying to estimate it for them and you'd like it as small as possible. In this module, we're not going to try to optimize any interval. But the choice of my cutoffs where I want to cut off the right area is going to be influenced by wanting to have the shortest interval. In the next video, we want to remove the ridiculous assumption that the variance is known for our population. We're going to introduce some new distributions that we're going to need in order to do this. I will see you there.
