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

Question

Scatter plot showing data points, a regression line, and annotations for explained variation and residuals.

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

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. 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 explained variation for this student? Awesome. So it appears for this particular problem we're asked to read off our multiple choice answers and we're asked to determine which of our statements made in our multiple choice answers best describes the explained variation for this particular student based on the information that is provided to us for the prom itself. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is the square distance between the student's predicted score and their actual score. I minus Y hat subscript I to the power 2. B is the square distance between the predicted score and the mean square parentheses Y hat subscript i minus y bar to the power 2. C is the square distance between the actual score and the mean square subscript i minus Y bar to the power 2 and finally D is the difference between the predicted score and the regression slope. Awesome. So with that in mind, let us recall that in regression analysis, we break down variation as follows. So I've provided a table that tells us the different types of variation, which we have 3 different types of variation in this table. We have total variation, explained variation, and unexplained variation. And based on these three types of variations, let's start with total variation. And our second column will have symbol slash formula, and then finally in our last column we'll have meaning. So focusing once again on total variation. The symbol is parentheses Y subscript I or Yi minus Y bar to the power 2. And this is the difference between the actual score and the average score. For explained variation, we have parentheses Y hat subscript I minus Y bar to the 2, and this is the difference between predicted score and the average score. And finally, the unexplained variation will be parenthesis Y subscript I minus Y hat subscript I to the power 2. And this is the difference between the actual score and the predicted score, AK also known as the residual. Fantastic. So let's quickly note that Y subscript I is the actual value, so this is the student's real score. So once again, this value of Y subscript I. Is the actual score of the student's real score score, whereas Yhat subscript I is the predicted value from the regression line and Y bar is the mean or the average score of all the students. So with that in mind, now that we've defined our variables, our first step that we need to take is we need to identify what explained variation means. So as we should recall and note, explained variation measures how far the predicted score is from the average score. So with that in mind, our second step now is we need to recall and use the following formula for explained variation. So for explained variation, which we'll call EV for short, so for explained variation, it is equal to parenthesisy hat subscript i minus Y bar to the power of 2. And as once again as we should recall, this is the squared distance between the predicted score and the mean score. So that will mean our final answer without a doubt based on the information we've gathered together, will have to be the square distance between the predicted score and the mean score will be parenthesisy hat subscript i minus Y bar to the 2. And this corresponds with multiple choice answer letter B, which once again is the square distance between the predicted score and the mean score is parenthesisy hat subscript i minus Y bar to the power of 2. And that's it. That is 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.

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

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

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"Graphical Analysis In Exercises 1–3, use the figure.2. Describe the explained variation about a regression line in words and in symbols."

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

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