A light source is placed at varying distances from a sensor. The intensity of light measured by the sensor is recorded. The results are as shown in the table below; the distance, \(x\), is in cm and the intensity of light, \(y\), is in lux.
| \(x\) | \(10\) | \(15\) | \(20\) | \(30\) | \(40\) | \(50\) | \(70\) | \(100\) |
|---|
| \(y\) | \(112.2\) | \(60.1\) | \(42.5\) | \(28.8\) | \(24.2\) | \(22.0\) | \(20.1\) | \(19.2\) |
Sketch the scatter diagram for the data. Hence, explain why a linear model \(y=a+bx\) is unlikely to be appropriate.[3]
The following two models are proposed, where \(c\), \(d\), \(e\) and \(f\) are constants.
\[y=c+\frac{d}{x}\qquad y=e+f\ln x\]
Calculate the product moment correlation coefficient for both models. State, with a reason, why the model \(y=c+\dfrac{d}{x}\) is the more appropriate model.[3]
Find the regression equation for the model \(y=c+\dfrac{d}{x}\), giving your constants \(c\) and \(d\) correct to \(2\) decimal places.[1]
Use your equation found in part (c) to estimate the light intensity when \(x=120\). Comment on the reliability of this estimate.[2]
In this context, what does the value of \(c\) in your regression line in part (c) represent?[1]
Suggest a physical interpretation for the constant \(c\).[1]