Suppose the regression line for a set of data, ŷ=3 x+b, passes through the point (2,5) . If x̅ a

step 1 So here we have the regression line for a set of data, which is y hat is equal to 3x plus b, and

Suppose the regression line for a set of data, ŷ=3 x+b,...

Suppose the regression line for a set of data, $\hat{y}=3 x+b$, passes through the point (2,5) . If $\bar{x}$ and $\bar{y}$ are the sample means of the $x$ - and $y$ -values, respectively, then $\bar{y}=$ (A) $\bar{x}$. (B) $\bar{x}-2 .$ (C) $\bar{x}+5$. (D) $\mu_{\bar{x}}$. (E) $3 \bar{x}-1$

Suppose the regression line for a set of data, $\hat{y}=3 x+b$, passes through the point (2,5) . If $\bar{x}$ and $\bar{y}$ are the sample means of the $x$ - and $y$ -values, respectively, then $\bar{y}=$
(A) $\bar{x}$.
(B) $\bar{x}-2 .$
(C) $\bar{x}+5$.
(D) $\mu_{\bar{x}}$.
(E) $3 \bar{x}-1$

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Key Concepts

Least Squares Regression

Least squares regression is a method used for modeling the relationship between a dependent variable and one or more independent variables by minimizing the sum of the squares of the differences between observed and predicted values. This method provides the best-fit line for the data in the sense that it minimizes these squared errors.

Regression Line Passing Through the Means

A key property of the least squares regression line is that it always passes through the point (x?, ?), where x? and ? represent the sample means of the independent and dependent variables, respectively. This property is crucial for understanding how the regression parameters relate to the central tendency of the data.

Slope-Intercept Form

The slope-intercept form of a line, expressed as y = mx + b, is commonly used in regression analysis, where m represents the slope and b represents the y-intercept. Given any point on the line, particularly (x?, ?), this form allows for the determination of the intercept when the slope is known.

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