In order to investigate whether there is a correlation between rainfall and crop yields, the total rainfall, $x$ mm, and the weights of a particular crop per square metre, $y$ kg, were recorded in a number of fields. The data are shown below.

$x$	36	72	44	74	64	50
$y$	2.2	8.4	1.8	7.4	4.3	2.2

1. Choose appropriate scales and label the $x$-axis and $y$-axis.

The $x$-values range from $36$ to $74$, so an appropriate scale for the horizontal axis is from $30$ to $80$. The $y$-values range from $1.8$ to $8.4$, so an appropriate scale for the vertical axis is from $0$ to $9$ or $10$. Label the horizontal axis as '$x$' and the vertical axis as '$y$'.

2. Plot the given data points accurately on the grid.

Plot the six coordinate points: $(36, 2.2)$, $(72, 8.4)$, $(44, 1.8)$, $(74, 7.4)$, $(64, 4.3)$, and $(50, 2.2)$.

1. Calculate the product moment correlation coefficient, $r$.

By entering the $x$ and $y$ values into a graphing calculator:
$r = 0.914098\dots \approx 0.914 \text{ (to 3 s.f.)}$

2. Explain the limitation of the correlation coefficient regarding the best model.

The value of $r$ only measures the strength of the linear relationship between $x$ and $y$. Even if $r$ is close to $1$, the points could lie closer to a curve, meaning a non-linear model might be a better fit.

1. Determine the product moment correlation coefficients (PMCC) for both models.

Using a graphing calculator to find the correlation for the model $y = a + bx$ (as calculated in part ii), the PMCC for the data pairs $(x, y)$ is:
$r_{xy} \approx 0.91409 \approx 0.914$

For the model $y = c + dx^2$, we calculate the PMCC for the data pairs $(x^2, y)$:
$r_{x^2y} \approx 0.93986 \approx 0.940$

2. Compare the PMCCs to determine the better model.

Since $|r_{x^2y}|$ is closer to $1$ than $|r_{xy}|$ ($0.940 > 0.914$), the points for $(x^2, y)$ lie closer to a straight line than the points for $(x, y)$.

Therefore, $y = c + dx^2$ is a better model.

1. Identify the appropriate model and find the equation of the regression line of $y$ on $x^2$.

From part (iii), the model $y = c + dx^2$ is the better fit because its product moment correlation coefficient ($r \approx 0.940$) is closer to $1$ than that of $y = a + bx$ ($r \approx 0.914$).
Using a graphing calculator to find the regression line of $y$ on $x^2$:
$y = 0.0015441x^2 - 0.88933$

2. Estimate the yield by substituting $x = 55$ into the regression equation.

Substitute $x = 55$ into the equation to find $y$:
$y = 0.0015441(55)^2 - 0.88933$
$y = 4.6709 - 0.88933$
$y \approx 3.78$

The estimated yield is $3.78\text{ kg/m}^2$.

3. Comment on the reliability of the estimation.

Since the rainfall $x = 55$ lies within the range of the given data ($36 \le x \le 74$), the estimate is found by interpolation. Together with the fact that the correlation is strong, the estimation is reliable.