2023 JPJC P2 Q9

The shear strength of soil is the ability of the soil to resist tensional stresses and sustain loads. The water content of soil is the ratio of the weight of water to the weight of the solids in a mass of soil. A soil investigation laboratory collected soil from a reclaimed site to study the effect of water content on shear strength. The information gathered about the shear strength \(P\), measured in KiloPascal (KPa), at different water content \(w\), measured in percentage is presented in the table.

1. Draw a scatter diagram of these data. Use your diagram to explain whether the relationship between \(P\) and \(w\) is likely to be well modelled by an equation of the form \(P=aw+b\), where \(a\) and \(b\) are onstants.[2]
2. By calculating the relevant product moment correlation coefficients, determine whether the relationship between \(P\) and \(w\) is modelled better by \(P=aw+b\) or \(P=a\sqrt{w}+b\). Explain how you decide which model is better, and state the equation in this case.[5]
3. Use your equation to estimate the shear strength when the water content is \(40%\). Explain whether your estimate is reliable.[2]
4. Subsequently, it is known that one data point is not presented in the information table and that the inclusion of this data point does not affect the appropriate equation in (ii).
   State the shear strength for the missing data point, giving your answer correct to the nearest integer.[1]
5. Given that \(1\) MegaPascal \(=1000\) KiloPascal, re-write your equation in (ii) so that it can be used when the shear strength is measured in MegaPascal. Explain whether the corresponding product moment correlation coefficient would differ from that calculated in (ii).[2]