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Conditions for inference on slope

Introducing the conditions for making a confidence interval or doing a test about slope in least-squares regression.

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Video transcript

- [Instructor] In a previous video, we began to think about how we can use a regression line and, in particular, the slope of a regression line based on sample data, how we can use that in order to make inference about the slope of the true population regression line. In this video, what we're going to think about, what are the conditions for inference when we're dealing with regression lines? And these are going to be, in some ways, similar to the conditions for inference that we thought about when we were doing hypothesis testing and confidence intervals for means and for proportions, but there's also going to be a few new conditions. So to help us remember these conditions, you might want to think about the LINER acronym, L-I-N-E-R. And if it isn't obvious to you, this almost is linear. Liner, if it had an A, it would be linear. And this is valuable because, remember, we're thinking about linear regression. So the L right over here actually does stand for linear. And here, the condition is, is that the actual relationship in the population between your x and y variables actually is a linear relationship, so actual linear relationship, relationship between, between x and y. Now, in a lot of cases, you might just have to assume that this is going to be the case when you see it on an exam, like an AP exam, for example. They might say, hey, assume this condition is met. Oftentimes, it'll say assume all of these conditions are met. They just want you to maybe know about these conditions. But this is something to think about. If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust. Now, the next one is one we have seen before when we're talking about general conditions for inference, and this is the independence, independence condition. And there's a couple of ways to think about it. Either individual observations are independent of each other. So you could be sampling with replacement. Or you could be thinking about your 10% rule, that we have done when we thought about the independence condition for proportions and for means, where we would need to feel confident that the size of our sample is no more than 10% of the size of the population. Now, the next one is the normal condition, which we have talked about when we were doing inference for proportions and for means. Although, it means something a little bit more sophisticated when we're dealing with a regression. The normal condition, and, once again, many times people just say assume it's been met. But let me actually draw a regression line, but do it with a little perspective, and I'm gonna add a third dimension. Let's say that's the x-axis, and let's say this is the y-axis. And the true population regression line looks like this. And so the normal condition tells us that, for any given x in the true population, the distribution of y's that you would expect is normal, is normal. So let me see if I can draw a normal distribution for the y's, given that x. So that would be that normal distribution there. And then let's say, for this x right over here, you would expect a normal distribution as well, so just like, just like this. So if we're given x, the distribution of y's should be normal. Once again, many times you'll just be told to assume that that has been met because it might, at least in an introductory statistics class, be a little bit hard to figure this out on your own. Now, the next condition is related to that, and this is the idea of having equal variance, equal variance. And that's just saying that each of these normal distributions should have the same spread for a given x. And so you could say equal variance, or you could even think about them having the equal standard deviation. So, for example, if, for a given x, let's say for this x, all of sudden, you had a much lower variance, made it look like this, then you would no longer meet your conditions for inference. Last, but not least, and this is one we've seen many times, this is the random condition. And this is that the data comes from a well-designed random sample or some type of randomized experiment. And this condition we have seen in every type of condition for inference that we have looked at so far. So I'll leave you there. It's good to know. It will show up on some exams. But many times, when it comes to problem solving, in an introductory statistics class, they will tell you, hey, just assume the conditions for inference have been met. Or what are the conditions for inference? But they're not going to actually make you prove, for example, the normal or the equal variance condition. That might be a bit much for an introductory statistics class.