\( \DeclareMathOperator{cosec}{cosec} \)

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The regression line of \( x \) on \( y \), often denoted as \( x = f(y) \), is a statistical tool used when \( y \) is considered the independent variable and \( x \) is the dependent variable. This is less common than the conventional regression of \( y \) on \( x \), but it is useful in situations where we are interested in predicting the value of \( x \) given \( y \). For example, one might use the regression of \( x \) on \( y \) if \( y \) is time and \( x \) is a process that unfolds over time.

The equation for the regression line of \( x \) on \( y \) is given by \( x = a + by \), where \( a \) represents the intercept and \( b \) the slope of the regression line. The slope indicates the change in \( x \) for a one-unit change in \( y \). It is important to note that this does not imply causation; rather, it is a way of describing how the two variables co-vary. To compute the coefficients \( a \) and \( b \), one would typically use the method of least squares, minimising the sum of the squares of the vertical distances of the points from the line - best done on a GDC.

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