"In Exercises 13-16, use the value of the correlation coeffici...

Question

"In Exercises 13-16, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?
15. r = 0.642"

15. r = 0.642"

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. If the correlation coefficient is R equals -0.523, what is the coefficient of determination? What proportion of the variation is explained and unexplained by the regression line? Awesome. So it appears for this particular prompt. We're asked to solve for the coefficient of determination. That is our first answer we're trying to solve for. And our 2nd and 3rd answers that we're trying to solve for is we're trying to determine what proportion of the variation is explained. That's our second answer, and our third answer is what proportion of the variation is unexplained. And both the explained and unexplained conditions are focused by the regression line. So once again, what is the proportion of the variation that is explained and unexplained by the regression line. So now that we know what we're ultimately trying to solve for, let's read off our multiple. multiple choice answers to see what our final answer set might be, noting that I will read the answer for the coefficient of determination first, then the answer for explained and the answer for unexplained, and note that the answers for explained and unexplained are both percentages. So A is 0.274 and 27.4 and 72.6. B is 0.523 and 52.3 and 47.7. C is 0.421 and 42.1 and 57.9. D is 62.5 and 37.5, and our coefficient of determination is 0.625 for D. So with that in mind, now that we know what our multiple choice answers are, our first step that we need to take in order to solve this particular problem is we need to calculate the coefficient of determination by recalling and using the following equation, which states that R squared. Is equal to 0. 523 to the power 2. is equal to, when we plug this into our calculator, 0.273529. Fantastic. Our second step now is we need to convert our coefficient of determination value into a percentage, so we need to take 0.273529, and we need to multiply it by 100. And when we do, and we round to one decimal place, we will find that it is approximately equal to 27.4%. And this will be our explained variation. So I'll call it EV explained variation. Fantastic, and that is one of our final answers. Hooray, we did it. So, and let us also note for our coefficient of determination, all of our multiple choice answers are rounded to three decimal places. So let us note that our coefficient of determination, as we discovered, will be equal to 0.274 when we round to three decimal places. So we could say that it's approximately equal to, once again, 0.274. Fantastic. So now, moving right along here, the coefficient of determination, as we should recall, tells us how much of the variation in the dependent variable Y can be explained by the independent variable X using the regression line. So as we should recall, the remaining variation is due to other factors not explained by the model. Such as randomness or other variables or noise. So with that in mind, our third step now is we need to calculate the unexplained variation. So we need to take 100% and subtract 27.4%, and when we perform that calculation, we will find that 72.6% is unexplained, which I'll call UE for short. And that is our 3rd and final answer. Hooray, we did it. So with that in mind, our final answer choice or multiple choice answer that has to be our correct and final answer has to be, without a doubt, multiple choice answer letter A. Which states that the coefficient of determination is 0.274 and 27.4% is explained, and 72.6% is unexplained. And that's it. That's our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

"In Exercises 13-16, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?
15. r = 0.642"

Key Concepts

Correlation Coefficient (r)

Coefficient of Determination (r^2)

Explained vs. Unexplained Variation

Watch next

"In Exercises 13-16, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?15. r = 0.642"

Key Concepts

Correlation Coefficient (r)

Coefficient of Determination (r^2)

Explained vs. Unexplained Variation

Watch next

"In Exercises 13-16, use the value of the correlation coefficient r to calculate the coefficient of determination r^2. What does this tell you about the explained variation of the data about the regression line? about the unexplained variation?
15. r = 0.642"