"In Exercises 7-12, match the description in the left column w...

Question

"In Exercises 7-12, match the description in the left column with its symbol(s) in the right column.

12. The point a regression line always passes through

a. \hat{y}_i

b. y_i

c. b

d. (\bar{x}, \bar{y})

e. m

f. \bar{y}"

Accepted Answer

Below 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 linear regression, the least squares regression line is the line that best fits the data by minimizing the square differences between observed and predicted values. One important property of this line is that it always passes through a specific point that represents the average of the data. Which of the following symbols represents the point that the regression line always passes through? Awesome. So it appears for this particular problem we're asked to look at our multiple choice answers, and we're asked to determine which of the following symbols from our multiple choice answers represents the point that the regression line always passes through. 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. Once again noting that they're all in parentheses. So A is Bhat subscript I, B is X I Y I C is M B. And D is X Y bar. Awesome. So, with that in mind, our first step is we need to recall and note that in simple linear regression, the least squares regression line has a guaranteed property. Meaning it always passes through the point whose coordinates are the mean of the X values and the mean of the y values and as we should recall, this point is written as parentheses X bary bar. In parentheses. OK, so let's repeat that because that's very important. So, as we should recall, in in simple linear regression, the least squares regression line has a guaranteed property. It is that it will always pass through the point whose coordinates are the mean of the X values and the mean of the of the Y values, which will mean that this point, as we should recall, is written as parentheses X Y bar, where X is the mean of the explanatory variable or the X values, and Y bar is the mean of the response variable or the Y values. So even though we've officially saw for a final answer, let's take a moment to briefly explain the other symbols and what they mean. So why hat subscript I, this is the predicted Y value for the I data point. Awesome. And why subscript I is the actual observed Y value for the data point. And B is the Y intercept of the regression line. So once again, B is the Y intercept of the regression line, and Y bar is the mean of all the Y values by itself, but it doesn't. But by itself, but it doesn't include the corresponding mean of X value. So once again, Y bar is the mean of all the Y values but by itself, meaning it does not include the corresponding mean of X values. So that means therefore, without a doubt. The point, the regression line always passes through is parentheses X, Y, and this corresponds to multiple choice answer letter D, which once again is parentheses X bar, 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 7-12, match the description in the left column with its symbol(s) in the right column.
12. The point a regression line always passes through
a. \hat{y}_i
b. y_i
c. b
d. (\bar{x}, \bar{y})
e. m
f. \bar{y}"

Key Concepts

Regression Line

Mean of x and y (Centroid)

Regression Line Equation Components

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