N2026 P2 Q8

Three researchers are investigating the effect of the amount of fertiliser used on crop yields.

1. Tang uses Fertiliser A. He calculates that the product moment correlation coefficient between the amount of fertiliser used per square metre and the crop yield per square metre is \(0.98\).
   State what else Tang should do before deciding whether a linear model is a good fit to the data.[1]
2. Xi uses Fertiliser B. He finds that the product moment correlation coefficient between the amount of fertiliser used per square metre and the crop yield per square metre is approximately \(0\).
   1. Explain how Xi should interpret the results of his investigation.[1]
   2. Sketch a scatter diagram with at least \(5\) points to illustrate a product moment correlation coefficient of approximately \(0\).[1]
3. Leia uses Fertiliser C. She sets up \(6\) test plots of land. The table shows the amount of fertiliser, \(x\) grams, used per square metre of each plot and the crop yield, \(y\) kilograms, per square metre of each plot.

   \(x\)	\(0\)	\(5\)	\(10\)	\(15\)	\(20\)	\(25\)
   \(y\)	\(7\)	\(14\)	\(16\)	\(21\)	\(21\)	\(20\)

   1. Sketch the scatter diagram for Leia's data, and explain how you know that the crop yield is not linearly related to the amount of fertiliser used.[2]
   2. By drawing sketches of the equations below, where \(c\), \(d\), \(e\) and \(f\) are positive constants, explain which is a better model for the relationship between the amount of fertiliser used and the crop yield.
      \[y=c\sqrt{x}+d\qquad\qquad y=ex^2+f\][3]
   3. Use the better model to find an estimate for the crop yield per square metre when the amount of fertiliser used is \(13\) grams per square metre. Give your answer correct to 3 significant figures.[3]