2025 DHS P2 Q9

1. The following scatter diagrams with \(5\) data points each have product moment correlation coefficients \(-0.7\), \(0\), \(1\), not necessarily in the given order. State the product moment correlation coefficients for the scatter diagrams I, II and III.[1]

2. A private developer hires workers for renovation works for its townhouse projects. The renovation works required for each townhouse in a certain project are identical. The private developer wishes to investigate how the number of working days taken to complete the renovation works for a townhouse for a certain project, \(y\), varies with the number of workers hired to renovate a townhouse, \(x\). The relevant data collected by the private developer is given in the following table.

1. Draw a scatter diagram for these values. Use your diagram to comment on whether a linear model would be appropriate to model the relationship between \(x\) and \(y\).[2]
2. By referring to the scatter diagram in part (b)(i) and the context of the question, choose and explain which one of the following models (A), (B) and (C) below, where \(a\) and \(b\) are positive constants, is the most appropriate for modelling the relationship between \(x\) and \(y\).[2]
   1. \(y=a-b\ln x\)
   2. \(y=a-bx^2\)
   3. \(y=a+\frac{b}{x^2}\)

Use the most appropriate model from part (b)(ii) for the rest of the question.

3. Find the equation for the relationship between \(x\) and \(y\), giving the numerical values of the coefficients and the product moment correlation coefficient for this model.[2]
4. Estimate the number of working days required to complete the renovation works for a townhouse when \(9\) workers are hired and comment on the reliability of your estimate.[2]
5. The private developer pays each worker a fixed amount of \(\$120\) for a day's work. Estimate the number of workers the developer should hire to minimise the total amount of workers' wages the developer will have to pay if he can only hire a maximum of \(20\) workers.[2]