一家小咖啡馆只销售一种咖啡。每年年初会根据市场情况对一杯咖啡的零售价进行评估和调整。根据 7 年收集的销售数据，咖啡馆老板郑先生（Mr Tay）研究了一杯咖啡的零售价 $x 对该年内平均每日售出杯数 $y$（杯）的影响。数据见下表。

$x$	2.0	2.2	2.5	3.0	4.0	4.4	4.5
$y$	280	250	190	150	90	100	70

1. Set up the axes for the scatter diagram.

Identify the variables to be plotted. The independent variable is the selling price $x$ (horizontal axis) and the dependent variable is the average number of cups $y$ (vertical axis). Ensure the axes on the provided grid cover the range of $x$ from at least $2.0$ to $4.5$ and $y$ from $0$ to $300$. Label the axes accordingly.

2. Plot the data points.

Plot each $(x, y)$ pair from the table onto the grid as a distinct point (e.g. using a cross '$\times$' or a dot).

The 7 points to plot are:
$(2.0, 280)$
$(2.2, 250)$
$(2.5, 190)$
$(3.0, 150)$
$(4.0, 90)$
$(4.4, 100)$
$(4.5, 70)$

1. Identify the anomalous point and state the action for the diagram

The data generally shows $y$ decreasing as $x$ increases. However, from $x = 4.0$ to $x = 4.4$, $y$ increases from $90$ to $100$ before dropping to $70$ at $x = 4.5$. Thus, $(4.4, 100)$ breaks the trend. The student should circle and label the point $(4.4, 100)$ as $P$ on their scatter diagram.

2. Explain why the remaining points fit the reciprocal model

For the remaining points, as $x$ increases, $y$ decreases, but the rate of decrease slows down (the gradient is negative but increasing towards zero). This decreasing, concave-upward shape matches the characteristics of the reciprocal graph $y = a + \frac{b}{x}$ (for $x > 0$ and $b > 0$).

1. Identify the valid data points and apply the transformation $X = \frac{1}{x}$.

From part (b), the incorrectly stated point $P$ is $(4.4, 100)$. We omit this point for our calculations.
Let $X = \frac{1}{x}$. The remaining pairs $(X, y)$ are:
$(0.5, 280), \left(\frac{5}{11}, 250\right), (0.4, 190), \left(\frac{1}{3}, 150\right), (0.25, 90), \left(\frac{2}{9}, 70\right)$

2. Use a graphing calculator to calculate the product moment correlation coefficient $r$ and the estimates for $a$ and $b$.

Entering the valid $(X, y)$ data into a graphing calculator for the linear regression model $y = a + bX$:
$r \approx 0.99711$
$a \approx -100.816$
$b \approx 756.863$

Rounding to 3 significant figures, we get:
$r = 0.997$
$a = -101$
$b = 757$

1. Estimate the value of $y$ at $x = 4.4$ using the regression model.

From part (c), we have $a \approx -100.817$ and $b \approx 756.863$.
Substitute $x = 4.4$ into the model $y = a + \frac{b}{x}$:
$y = -100.817 + \frac{756.863}{4.4}$
$y \approx 71.197 \approx 71.2 \text{ (to 3 s.f.)}$

2. State two reasons why the estimate is expected to be reliable.

Reason 1: The estimated value is found by interpolation, since $x=4.4$ lies well within the given data range of $2.0 \le x \le 4.5$.
Reason 2: The product moment correlation coefficient for the transformed model ($r \approx 0.997$) is very close to $1$, indicating a very strong positive linear correlation between $y$ and $\frac{1}{x}$, so the model fits the data extremely well.