IB Mathematics SL 2.5 Linear models AI SL Paper 1- Exam Style Questions- New Syllabus
Question
| Number of Passengers \(x\) | 1 | 2 | 4 | 5 | 7 | 15 |
|---|---|---|---|---|---|---|
| Recommended Minimum Volume \(y\) | 1000 | 1350 | 2225 | 2500 | 3150 | 5800 |
(ii) State what the value of \(a\) represents in the context of this problem.
Most-appropriate topic codes (IB Mathematics: applications and interpretation SL guide):
• SL 2.5: Modelling with linear models — part (b)(ii)
• SL 2.6: Modelling skills: reading, interpreting and making predictions based on the model — part (c)
▶️ Answer/Explanation
(a)
Enter the data into a graphic display calculator and perform linear regression analysis to obtain the Pearson correlation coefficient \(r\).
\(r = 0.999\) (to three significant figures) or \(0.999\) (from calculator output \(0.998886…\)).
\( \boxed{r = 0.999} \)
(b) (i)
Use the calculator to perform linear regression (\(y\) on \(x\)).
\(y = 340x + 742\) (e.g., \(y = 340.379…x + 742.015…\)).
\( \boxed{y = 340x + 742} \)
(b) (ii)
In the regression equation \(y = ax + b\), the coefficient \(a\) represents the slope. In this context, it indicates the average increase in the recommended minimum volume of the balloon per additional passenger.
For each additional passenger, the recommended minimum volume increases by approximately 340 cubic metres.
\( \boxed{\text{The volume increases by about } 340 \text{ m}^3 \text{ per passenger.}} \)
(c)
Substitute \(x = 10\) into the regression equation \(y = 340x + 742\).
\( y = 340 \times 10 + 742 = 3400 + 742 = 4142 \)
The recommended minimum volume is 4140 cubic metres (or 4142 m³ depending on calculation precision).
\( \boxed{4140 \text{ m}^3} \)