Final answer:
In simple linear regression, the formula for the slope (β1) and intercept (β0) are calculated using the summations of x and y variables based on the least-squares method. The slope (β1) is not confined to the range -1 to 1; rather, this range applies to the correlation coefficient (r), implicating the strength of a linear relationship. Outliers can greatly influence the regression equation.
Step-by-step explanation:
In simple linear regression, the model is given by y = β0 + β1x + ε, where β0 is the y-intercept, β1 is the slope of the regression line, and ε represents the error term.
To derive the formula for the slope (β1) and intercept (β0), we use the method of least squares, which minimizes the sum of the squared deviations between the actual values and the values predicted by the regression line.
The formula for the slope (β1) is:
β1 = Σ(n)(xiyi) - (Σ(xi)Σ(yi)) / (Σ(n)(xi^2) - (Σ(xi))^2)
And the y-intercept (β0) is calculated using the formula:
β0 = Σ(yi)/n - β1(Σ(xi)/n)
Once you have these coefficients, you can create the regression line for a set of data points. The coefficient (β1) indeed ranges between -1 and 1, indicating the strength of the linear relationship between the variables x and y. However, it is actually the correlation coefficient (r) that must lie within this range, not the slope (β1) of the regression line.
Also, caution must be taken when predicting values outside the range of observed x values, as this can lead to extrapolation, which may be unreliable. An outlier is an observation that does not fit the rest of the data and can significantly affect the slope and intercept of a regression line.