What does it mean to say that the linear correlation coefficie...

Question

What does it mean to say that the linear correlation coefficient between two variables equals 1? What would the scatter diagram look like?

Accepted Answer

Welcome back, everyone. In this problem, which description best matches a sample correlation coefficient of R equals 0? A says it means that there is a perfect positive linear association and all points lie on a line. B says there is necessarily independence between the variables. C says there is no linear association, but a nonlinear association may still exist, and the D says the scatter plot must be symmetric about the origin. Now what do we know about this scenario where our sample correlation coefficient is equal to zero? Well, when that happens, it indicates there is no linear relationship. Between the two variables, because remember, our correlation coefficient tells us how or gives us a degree uh to the level of how two variables are related. OK. So if it's zero, that just means there's no relationship between the variables. But this does not rule out the possibility of a nonlinear relationship, right? For example, the variables could be related in a curvilinear fashion, such as a quadratic or exponential relationship which a linear correlation coefficient would fail to detect. In other words, When R equals 0, it doesn't it, it tells us that this isn't there's no linear relationship, so there's nothing like this, but. It could be the case where we could have some relation like this. I doesn't tell us anything about that scenario. So now that we know that, let's try to figure out which of these options are correct. In A says there's a perfect positive linear association and all points lie on a line, but that's incorrect because like we said, a per there is no when R equals 0, sorry, there is no linear association between the variables. A perfect positive linear association corresponds to a value of R being equal to 1, like what is in our sketch here. So this would be a scenario for a. In answer choice B, it says there's necessarily independence between the variables, but a correlation of R equals 0 indicates no linear relationship. But independence means that the variables have no statistical relationship at all, and that includes nonlinear relationships. So a correlation of zero does not imply that the variables are independent. Now in answer try C, it says there's no linear association, but a nonlinear association may still exist, a scenario we discussed earlier. So that means C is the correct answer. And we can be sure we're right, because if we look at answer twice D where it says the scatter plot must be symmetric about the origin, this correlation coefficient R equals 0 does not require the scatter plot to be symmetric around the origin. It could take many shapes and symmetry is not a requirement for R to be equal to 0. Therefore, C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

What does it mean to say that the linear correlation coefficient between two variables equals 1? What would the scatter diagram look like?

Key Concepts

Linear Correlation Coefficient

Interpretation of r = 1

Scatter Diagram Representation

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