Transform variables and compare models

Transform units and track the sign of correlation

Write the variable transformation explicitly. For correlation, check the signs of both scale factors. For a transformed regression, identify the scale on which residuals were minimised, then undo the transformation to interpret the prediction.

For defined correlation and nonzero scale factors, if $X=a+bU$ and $Y=c+dV$, then

$$r_{XY}=\operatorname{sgn}(bd)\,r_{UV}.$$

Translations do not change correlation. Positive rescaling of either variable does not change it; reversing just one axis changes its sign. Reversing both leaves the sign unchanged. Scaling a variable by zero makes it constant, so correlation is undefined.

The source introduces $p=0.3x+7$ and $q=-4y-50$, then says such linear transformations do not affect correlation. For this particular pair, the product of scale factors is negative, so $r_{pq}=-r_{xy}$: the magnitude stays the same but the sign reverses. Its later precise condition $bd>0$ is the condition for an unchanged value; the general “multiply or divide” wording must retain that condition.

Track what changes in the fitted line

For regression, translating data preserves the gradient but can change the intercept. Rescaling axes generally changes the gradient as well. In Example 9, the scales are both positive, so the two calculated coefficients really are equal. The source notes that a covariance proof lies beyond the syllabus; the sign rule here makes the needed qualification explicit without requiring that proof.

Linearise a nonlinear relationship

A nonlinear pattern can sometimes be represented by a straight line after transforming variables. The source introduces quadratic, logarithmic and reciprocal relationships; fit the response against the correct transformed explanatory variable and inspect the transformed scatter too. Logarithms require positive arguments, and reciprocals require nonzero inputs.

A straight-line trend changes at a constant rate. Curvature may show an increasing or decreasing rate of change, but that description alone does not identify an exponential model. The source’s broad exponential wording is therefore a description to check against the actual proposed formula. Example 10 fits logarithms of both variables, giving a power law; Example 11 compares speed and squared speed as explanatory inputs.

Return a transformed fit to the measurement scale

Positive observations at explanatory values $0\le x\le4$ give the fitted model $\ln y=1+0.5x$. Write the corresponding y model, estimate y at x=2 and interpret a one-unit increase in x. Discuss prediction at x=8 and explain which residuals were minimised by fitting the stated line.

After your attempt: compare the reasoning

Exponentiating gives $y=e\,e^{0.5x}$, so the model prediction at x=2 is $e^2\approx7.3891$. A one-unit increase in x multiplies the fitted y value by $e^{0.5}\approx1.6487$; it does not add 0.5 to y. The logarithm requires positive y values.

The x=8 prediction is outside the observed explanatory range and needs additional justification for the continued model shape. The fitted line minimises squared residuals in $\ln y$, not in y on its original scale. A good transformed correlation alone does not establish that every original-scale prediction is reliable.