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. 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.

x	3	5	8	10	13	15	18	20
y	92	50	33	25	21	19	17	16

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 $y$. 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 $y$. (A) $y = a - b\ln x$ (B) $y = a - b x^{2}$ (C) $y = a + \dfrac{b}{x^{2}}$ Use the most appropriate model from part (b)(ii) for the rest of the question. 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. 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. 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.

1. Match each scatter diagram to its corresponding product moment correlation coefficient based on the linear trends.

Scatter Diagram I exhibits no clear linear trend, so its product moment correlation coefficient is $0$.

Scatter Diagram II shows points that are collinear on an increasing straight line, which indicates a perfect positive linear relationship, so its product moment correlation coefficient is $1$.

Scatter Diagram III shows a decreasing trend but the points are not perfectly collinear, indicating a moderately strong negative linear relationship, so its product moment correlation coefficient is $-0.7$.

1. Draw the scatter diagram by plotting the given data points.

Set up a Cartesian coordinate system with $x$ on the horizontal axis and $y$ on the vertical axis. Accurately plot the points $(3, 92)$, $(5, 50)$, $(8, 33)$, $(10, 25)$, $(13, 21)$, $(15, 19)$, $(18, 17)$, and $(20, 16)$.

2. Comment on the appropriateness of a linear model.

Observe the plotted points. The pattern formed by the points shows a steep decrease initially which gradually flattens out, clearly forming a curve rather than a straight line. Therefore, a linear model is not appropriate.

1. Reject model (B) based on the shape of the scatter diagram

From the scatter diagram, $y$ decreases at a decreasing rate (the curve is concave upwards) as $x$ increases. For model (B) $y = a - bx^2$, $y$ would decrease at an increasing rate as $x$ increases. Thus, model (B) is not appropriate.

2. Reject model (A) based on the problem context

In the given context, the number of working days $y$ cannot be negative. For model (A) $y = a - b\ln x$, as the number of workers $x$ gets very large, the term $-b\ln x$ will eventually cause $y$ to become negative (since $a$ and $b$ are positive constants). Thus, model (A) is not appropriate.

3. Validate and select model (C)

For model (C) $y = a + \dfrac{b}{x^2}$, since $a, b > 0$, $y$ is always positive for $x > 0$. Also, as $x$ increases, $y$ decreases at a decreasing rate and approaches a positive constant $a$. This fits both the shape of the scatter diagram and the physical context (a minimum base amount of time is required to complete the renovation). Therefore, model (C) is the most appropriate.

1. Identify the model chosen in part (b)(ii) and the appropriate transformation of variables.

From part (b)(ii), the most appropriate model is (C) $y = a + \frac{b}{x^2}$ (since as $x \to \infty$, $y$ approaches a positive constant $a$, which is physically meaningful for the minimum number of days required; models (A) and (B) would output negative days for large $x$).

We thus use the transformed variables $X = \frac{1}{x^2}$ and $Y = y$.

2. Calculate the regression coefficients and the product moment correlation coefficient using a graphing calculator.

Entering the data points $\left(\frac{1}{x^2}, y\right)$ into a graphing calculator and performing a linear regression of $Y$ on $X$, we obtain:

$a \approx 17.488$
$b \approx 690.75$
$r \approx 0.99239$

Rounding the coefficients to 3 significant figures, we get $a = 17.5$ and $b = 691$.

The equation is $y = 17.5 + \frac{691}{x^2}$ and the product moment correlation coefficient is $r = 0.992$ (to 3 s.f.).

1. Use the regression equation derived in part (b)(iii) to estimate the number of days for $x=9$.

From the data, using the linear model $y = a + \frac{b}{x^2}$, we can find the regression equation of $y$ on $\frac{1}{x^2}$ to be:
$y = 17.488 + \frac{690.74}{x^2}$
Substitute $x = 9$ into the equation:
$y = 17.488 + \frac{690.74}{9^2}$
$y = 26.015 \approx 26.0$

2. Comment on the reliability of the estimate.

The estimate is reliable because $x = 9$ is within the range of the given data values for $x$ ($3 \le x \le 20$), so this is an interpolation. Furthermore, the product moment correlation coefficient for this model is $r \approx 0.992$, which is very close to $1$, indicating a strong linear relationship.

1. Formulate the total wage cost function using the regression model established in earlier parts.

From part (b)(iii), the most appropriate regression model is (C) with equation $y = 17.488 + \dfrac{690.75}{x^2}$.
Let the total wage cost be $C$.
The developer pays each of the $x$ workers $120 per day for $y$ days, so:
$C = 120xy = 120x \left( 17.488 + \dfrac{690.75}{x^2} \right)$
$C = 2098.6x + \dfrac{82890}{x}$

2. Find the optimal continuous value of $x$ that minimises the cost function.

To find the minimum cost, we set $\dfrac{\mathrm{d}C}{\mathrm{d}x} = 0$:
$\dfrac{\mathrm{d}C}{\mathrm{d}x} = 2098.6 - \dfrac{82890}{x^2} = 0$
$x^2 = \dfrac{82890}{2098.6} \approx 39.498$
$x \approx 6.2847$
(Note: Students may also use a graphing calculator to locate this minimum.)
The second derivative $\dfrac{\mathrm{d}^2C}{\mathrm{d}x^2} = \dfrac{165780}{x^3} > 0$ for $x > 0$ confirms this is a minimum point.

3. Test the nearest integer values of $x$ to determine the discrete minimum within the given constraint.

Since the number of workers $x$ must be a positive integer and is subject to the maximum limit of 20, we test the integers adjacent to $6.2847$:
When $x = 6$, $C = 2098.6(6) + \dfrac{82890}{6} \approx 26406.60$
When $x = 7$, $C = 2098.6(7) + \dfrac{82890}{7} \approx 26531.60$

Comparing the costs, the total wages paid are minimised when $x = 6$.