"Graphical Analysis In Exercises 1–3, use the figure.

Question

Scatter plot showing data points, a regression line, and annotations for residuals and average values.

Describe the unexplained variation about a regression line in words and in symbols."

Accepted Answer

All right, hello, everyone. So, this question says, in a study examining the relationship between study hours and test scores, a linear regression line was fitted to the data. One student studied for 6 hours and scored 82, while the average test score of all students was 75. Which of the following best describes the unexplained variation for this student? And here we have 4 different answer choices labeled A through D. All right, so first and foremost, in regression analysis, we talk about variation in a few different ways. The first way is total variation. Which is expressed as Y1 subtracted by y bar squared. Now, what total variation tells you. Is simply the difference between the actual score and the average score. For that group But there's also both explained and unexplained variation. So the explained variation. Is represented symbolically. By why hat. So 1 subtracted by Y bar squared. Now, this instead tells you the difference between the predicted score, based on the regression line, subtracted by the average score. Oops. Did not mean to do that. And you also have, as mentioned before, the unexplained variation. Which is Represented by Y sub 1 subtracted by Y hat. Sweared Now the unexplained variation refers to the difference between the actual score. Subtracted by the predicted score. But just so we're extra clear here, let's clarify what these variables mean in this context, right? Why in this case refers to the actual value or the score that the student actually received. Why hat? Represents the predicted value obtained from the linear regression line. And why bar represents the mean. Of all the students in the group. So the unexplained variation can be thought of as the leftover error. It's the difference between the actual score and what the regression model predicted. Now, the difference between actual and predicted is referred to the residual, or is referred to as the residual. So the unexplained variation is the residual squared. It represents the part of the score that is not explained by the regression model. So going back to our answer choices. Our correct answer is going to be option A in the multiple choice, which reads the squared distance between the student's predicted score and their actual score, or why one subtracted by Y hat one. Squared And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

"Graphical Analysis In Exercises 1–3, use the figure.

Describe the unexplained variation about a regression line in words and in symbols."

Key Concepts

Regression Line

Residuals

Unexplained Variation

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