科学家们正在研究一种特定竹子的生长情况。他们希望了解竹茎的高度随时间的变化。下表显示了竹茎在发芽后年龄为 $t$ 周时的绝对高度 $h$ 英寸。

$t$	1	2	3	4	5	7	8	9
$h$	3.6	7.4	10.2	11.7	12.6	14.1	14.5	15.5

1. Draw the scatter diagram on a coordinate grid.

Set up a Cartesian plane with the horizontal axis as $t$ (from 0 to 10) and the vertical axis as $h$ (from 0 to 16). Plot the coordinates $(1, 3.6)$, $(2, 7.4)$, $(3, 10.2)$, $(4, 11.7)$, $(5, 12.6)$, $(7, 14.1)$, $(8, 14.5)$, and $(9, 15.5)$.

2. Explain why $h = a + bt$ is not a suitable model.

From the scatter diagram, the plotted points form a curve where the gradient decreases as $t$ increases. The equation $h = a + bt$ represents a straight line graph with a constant gradient (rate of increase). Since the points do not follow a linear trend, this model is not suitable.

3. Explain why $h = a + bt^2$ is not a suitable model.

The equation $h = a + bt^2$ (with $b > 0$) implies an increasing gradient, where $h$ would increase at an increasing rate (a curve that opens upwards). However, the scatter diagram clearly shows $h$ increasing at a decreasing rate (a curve that is concave downwards). Thus, this model is also not suitable.

1. Calculate the product moment correlation coefficient ($r$) for both models using a graphing calculator.

Let $r_A$ and $r_B$ be the product moment correlation coefficients for Model A and Model B respectively.
Using a graphing calculator:
For Model A ($h$ on $\sqrt{t}$): $r_A \approx 0.9736$
For Model B ($h$ on $\ln t$): $r_B \approx 0.9966$

2. Compare the correlation coefficients to determine which model is a better fit.

Since the value of $|r_B|$ is closer to $1$ than $|r_A|$, Model B provides a stronger linear correlation.
Therefore, Model B is a better fit to the data.

3. State the equation of the least squares regression line for the chosen model to 3 decimal places.

For Model B ($h$ on $\ln t$), the graphing calculator gives the regression line parameters:
$c \approx 3.89017$
$d \approx 5.31137$
Rounding to 3 decimal places, the equation is:
$h = 3.890 + 5.311\ln t$

1. Obtain the regression line from (b) and substitute $t = 6$ to estimate the height.

From (b), using a graphing calculator to find the correlation coefficients for the two models:

• Model A ($h$ on $\sqrt{t}$): $r \approx 0.9735$
• Model B ($h$ on $\ln t$): $r \approx 0.9966$

Since $|r|$ is closer to $1$ for Model B, it is the better fit. The least squares regression line (given to 3 d.p. as required in (b)) is $h = 3.890 + 5.311 \ln t$.

Substitute $t = 6$ to estimate the length:
$h = 3.890 + 5.311 \ln 6$
$h \approx 13.406$

The estimated length of a 6-week-old bamboo stalk is $13.4$ inches (to 3 s.f.).

2. Comment on the reliability of the estimate.

The estimate is reliable because $t = 6$ lies within the range of the given data ($1 \le t \le 9$), meaning this is an interpolation, and the chosen model has a strong correlation to the data ($r \approx 0.997$).

1. Comment on the long-term behaviour of the model compared to the real-world context

As $t \to \infty$, $h \to \infty$. According to the model, the height of the bamboo stalk would increase indefinitely to infinity, which is impossible in reality as it will eventually reach a maximum height. Thus, the model is not suitable in the long run.

1. Formulate the expression for the sum of squared residuals using the previously obtained regression line.

$S = \sum (h - H)^2$
Using the least squares regression line obtained in (b), where $H = 3.890 + 5.311\ln t$, the sum required is the residual sum of squares.

2. Evaluate the sum using a graphing calculator.

Using the graphing calculator's statistical operations to find the sum of squared residuals for the regression of $h$ on $\ln t$:
$S \approx 0.7732$
$S = 0.77 \text{ (to 2 d.p.)}$

1. State the least squares property to compare the sum of squared residuals.

The least squares regression line $H = c + d\ln t$ is mathematically derived to minimise the sum of the squares of the residuals, $S = \sum (h - H)^2$. For any other arbitrary constants $p$ and $q$ used in a different model $H' = p + q\ln t$, the sum of squared residuals $\sum (h - H')^2$ must be greater than (or equal to) the minimum value $S$.