The number of years ($x$) an employee has worked for the company and the corresponding salary increment, in dollars ($y$), received by the employee are given in the table.

Years, $x$	6	7	9	11	13	15	18
Amount, $y$	155	170	211	230	248	260	265

1. Set up and label the axes with appropriate scales

Draw a horizontal axis labelled 'Years, $x$' covering values from at least $6$ to $18$, and a vertical axis labelled 'Amount, $y$' covering values from at least $155$ to $265$. A scale break may be used on the $y$-axis.

2. Plot the given data points

Accurately plot the following coordinates on the grid:
$(6, 155)$
$(7, 170)$
$(9, 211)$
$(11, 230)$
$(13, 248)$
$(15, 260)$
$(18, 265)$

1. Find the product moment correlation coefficient for $x$ and $y$.

By entering the given values of $x$ and $y$ into the statistics mode of a graphing calculator,
$r = 0.94827\dots$
$r \approx 0.9483$ (correct to 4 d.p.)

2. Find the product moment correlation coefficient for $\ln x$ and $y$.

By applying the natural logarithm to the values of $x$, enter the paired values of $\ln x$ and $y$ into the graphing calculator,
$r = 0.98494\dots$
$r \approx 0.9849$ (correct to 4 d.p.)

1. Compare the product moment correlation coefficients calculated in part (ii).

Calculations from part (ii):
Using the data table and a calculator, the product moment correlation coefficients are:

• For $x$ and $y$: $r \approx 0.9483$
• For $\ln x$ and $y$: $r \approx 0.9849$

The correlation coefficient $r$ for $\ln x$ and $y$ is closer to $1$ than that for $x$ and $y$, indicating a stronger positive linear relationship between $\ln x$ and $y$.

2. Analyse the shape of the scatter diagram from part (i) to conclude the best model.

Observation from part (i):
Based on the table of values, plotting $y$ against $x$ shows that as $x$ increases, the rate of increase of $y$ visibly decreases (the trend curves downwards rather than maintaining a constant gradient). A logarithmic function models this shape much better than a straight line.

Since both the scatter diagram and the correlation coefficients support a non-linear log relationship, $y = c \ln x + d$ is the better model.

1. State the relationship between the variables and explain why the given regression lines are inappropriate

The number of years, $x$, is the independent (or controlled) variable, and the salary increment, $y$, is the dependent variable. Therefore, the regression line of the dependent variable on the independent variable (i.e. $y$ on $x$, or $y$ on $\ln x$) must be used, even when estimating $x$ from $y$.

1. Identify and find the equation of the suitable regression line.

Since $x$ (years) is the predetermined independent variable, we must regress the dependent variable $y$ on $x$. Based on part (iii), the logarithmic model is the better fit. Therefore, the suitable regression line is the regression line of $y$ on $\ln x$.

Using a graphing calculator with the given data points:
$y = 106.559\ln x - 31.263$
$y = 107 \ln x - 31.3 \text{ (to 3 s.f.)}$

2. Use the equation to find the required estimate for $x$.

Substitute $y = 200$ into the unrounded regression equation:
$200 = 106.559 \ln x - 31.263$
$106.559 \ln x = 231.263$
$\ln x \approx 2.1703$
$x = e^{2.1703} \approx 8.76 \text{ (to 3 s.f.)}$

3. Comment on the reliability of the estimate.

The value $y = 200$ falls within the given data range for $y$ ($155 \le y \le 265$), meaning this is an interpolation. Furthermore, the correlation coefficient $r \approx 0.985$ is close to 1, indicating a strong fit for the model. Therefore, the estimate is reliable.