A study was conducted on a group of pre-diabetic seniors to examine whether the number of hours of exercise per week ($t$) has any effect on their fasting blood sugar levels ($S$, measured in mg/dL). The results were as follows.

$t$	1.5	1.8	2.1	2.4	2.5	3.1	3.5	4.2
$S$	119	108	100	95	93	88	86	85

1. Draw and label the axes with appropriate scales.

Draw the horizontal axis labelled $t$ (hours per week) with a scale covering $0$ to $4.5$.
Draw the vertical axis labelled $S$ (mg/dL) with a scale covering at least $80$ to $125$.

2. Plot the given data points accurately on the scatter diagram.

Plot the $8$ coordinates exactly as given in the table:
$(1.5, 119), (1.8, 108), (2.1, 100), (2.4, 95), (2.5, 93), (3.1, 88), (3.5, 86), (4.2, 85)$

1. Comment on the appropriateness using the scatter diagram.

From the scatter diagram, the plotted points do not lie close to a straight line. Instead, they follow a curve where the rate of decrease of $S$ slows down as $t$ increases.

2. Comment on the appropriateness using the context of the question.

In the context of the question, if a linear model with a negative gradient were used, the fasting blood sugar level $S$ would continue to decrease at a constant rate and eventually become negative for large values of exercise hours $t$. Since blood sugar levels cannot be negative and must taper off to a safe, positive baseline, a linear model is not appropriate.

1. Observe the trend in the provided data.

From the data points provided in the table, as $t$ increases, $S$ decreases.

2. Evaluate the shapes of the proposed models given that $a$ and $b$ are positive.

For $t > 0$ and $b > 0$:

• Model A ($S = a + bt^2$) represents a positive quadratic curve, meaning $S$ increases as $t$ increases.
• Model C ($S = a + b \ln t$) represents a logarithmic curve, meaning $S$ also increases as $t$ increases.
• Model B ($S = a + \frac{b}{t}$) represents a hyperbola where $S$ decreases as $t$ increases.

3. Conclude the most appropriate model based on the matching trend.

Since the scatter diagram (and data) shows a decreasing trend, Model B is the most appropriate as it is the only model that yields a decreasing graph.

1. Identify the transformed variables from the appropriate model deduced in part (c).

From the data, $S$ decreases as $t$ increases. Since $b > 0$, the only appropriate model from part (c) that results in a decreasing relationship is $\textbf{B: } S = a + \frac{b}{t}$, because $\frac{1}{t}$ decreases as $t$ increases.

The linear relationship is between $\frac{1}{t}$ and $S$.

2. Compute the product moment correlation coefficient using a graphing calculator.

Using a graphing calculator, input the data pairs $\left(\frac{1}{t}, S\right)$:
$\left(\frac{1}{1.5}, 119\right), \left(\frac{1}{1.8}, 108\right), \dots, \left(\frac{1}{4.2}, 85\right)$

$r = 0.988065\dots \approx 0.988 \text{ (to 3 s.f.)}$

1. Find the equation of the regression line based on the chosen model.

From part (c), Model B ($S = a + \frac{b}{t}$) is the most appropriate as the blood sugar $S$ decreases at a decreasing rate for increasing positive values of $t$, and $a, b > 0$.
Using a graphing calculator with $X = \frac{1}{t}$ and $Y = S$:
$a = 62.438$
$b = 81.661$
The regression equation is $S = 62.4 + \frac{81.7}{t}$ (to 3 s.f.).

2. Estimate the exercise time for a blood sugar level of $99$ mg/dL.

Substitute $S = 99$ into the regression equation:
$99 = 62.438 + \frac{81.661}{t}$
$36.562 = \frac{81.661}{t}$
$t = \frac{81.661}{36.562} \approx 2.2335$
The estimated exercise time is $2.23$ hours (to 3 s.f.).

3. Comment on the reliability of the estimate.

The estimate is reliable because $S = 99$ lies well within the collected data range for fasting blood sugar levels ($85 \le S \le 119$), which means the prediction involves interpolation.