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What is the formula for the Least Squares Regression Line (LSRL)?
$\hat{y} = a + bx$
In the LSRL formula, $\hat{y} = a + bx$, what does 'a' represent?
The y-intercept (where the line crosses the y-axis).
In the LSRL formula, $\hat{y} = a + bx$, what does 'b' represent?
The slope (how much ŷ changes for every one-unit increase in x).
Explain the concept of linear regression.
Linear regression models the relationship between an explanatory variable (x) and a response variable (y) using a straight line.
Explain the concept of the Least Squares Regression Line (LSRL).
It's the line that minimizes the sum of the squared differences between the actual y-values and the predicted y-values (ŷ).
Explain why extrapolation can be risky.
Extrapolation assumes the linear trend continues beyond the observed data range, which may not be true, leading to unreliable or nonsensical predictions.
What is the definition of explanatory variable?
A variable (x) used to predict or explain the value of the response variable.
What is the definition of response variable?
A variable (y) that is predicted or explained by the explanatory variable.
What is the definition of Least Squares Regression Line (LSRL)?
The line that minimizes the sum of the squared differences between observed and predicted y-values.
What does ŷ represent in the LSRL equation?
The predicted value of the response variable.
What is extrapolation?
Using the LSRL to predict values outside the range of the original data.