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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 11 · 11.12 Mini IA — DP exploration practice

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze
1.Dependent: TT; Independent: hh.
2.CHF 3 per item; fixed cost CHF 5
3.85% of the variation in the dependent variable is explained by the model.
4.9
5.70
6.Positive (direct) correlation.
7.Quadratic.
8.(i) observed − predicted
9.Logistic model (or saturating curve).
10."How does caffeine intake (mg) affect reaction time (ms) in young adults?"
Silver
11.y=3x−1y = 3x - 1
12.CHF 170
13.(a) Interpolation; (b) Extrapolation
14.Quadratic, because r2r^2 is much higher and gravity gives quadratic motion.
15.1
16.−4.9-4.9: half of gravitational acceleration (= −12g-\tfrac{1}{2}g); 20: initial velocity (m/s); 1: initial height (m).
17.Most values lie within ~3 of 12 (i.e. between 9 and 15).
18.CHF per item
19.Only for t=0t = 0 (where P=1P = 1, the maximum). For t>0t > 0 the model gives P>1P > 1, which is impossible.
20.r2=0.64r^2 = 0.64; 64% of variation explained.
Gold
21.(a) Each CHF 1 of advertising → CHF 1.80 extra revenue. (b) 84% of variation in revenue explained by advertising.
22.(a) Weak fit (only 38% explained); (b) Negative gradient means more sleep → faster reaction; plausible direction.
23.Linear is preferable — extra parameter of quadratic barely improves r2r^2. (Occam.)
24.(1) There is a theoretical reason to expect the model holds beyond observed range; (2) Sensible behaviour at x=50x = 50 (no breakdown).
25.CHF 250 = expected revenue with zero advertising spend (baseline). Plausible if there's organic demand.
26.y=3x+5y = 3x + 5
27.(a) 50 (thousand); (b) ~4% per year
28.(i) Larger sample; (ii) Random assignment of music vs no-music; (iii) Repeated trials.
29.The point is an outlier — its yy-value is far above the predicted 19. Investigate the cause (data entry error? unusual circumstance?).
30.No — correlation does not imply causation. Both are driven by temperature (a confounding variable).
Platinum
31.(a) Quadratic. (b) 2.78 m. (c) t≈1.27t \approx 1.27 s.
32.(a) Slope decreases (less pulled up by outlier). (b) r2r^2 increases (less noise). (c) Intercept rises slightly.
33.y=2x+2.33y = 2x + 2.33 approx
34.(a) Model B. (b) Caffeine improves reactions up to some optimum, then worsens (jittery). (c) Extrapolation beyond data range is risky.
35.(a) "Does the angle of release predict free-throw success at distance 4.6 m?" (b) Variables: release angle (degrees), success (yes/no), release speed. (c) Logistic regression / probability tree; quadratic motion equation.
36.a=5a = 5, b=2b = 2
37.(a) 1000; (b) 5% per year; (c) ≈13.9\approx 13.9 years
38.(a) Predicts 70 cm at birth, much higher than typical (~50 cm); growth rate non-constant. (b) Growth slows / stops; the linear model overestimates.
39.(a) Moderate (negative). (b) r2=0.16r^2 = 0.16 → 16% of variation explained. (c) Age, screen-time policy, household routine.
40.(a) "More sleep the night before → higher quiz score." (b) (i) collect 30 participants; (ii) record hours of sleep night before; (iii) take 20-question quiz; (iv) score quiz; (v) regress score on sleep. (c) "A non-significant correlation or one in the opposite direction would reject."

Pack B — Answers

Bronze
1.Dependent: CC; Independent: nn.
2.CHF 4 per item; fixed cost CHF 2
3.30% of variation is explained — fit is weak.
4.7
5.11
6.Negative (inverse) correlation.
7.Exponential.
8.(i) observed − predicted
9.Linear.
10."How does the number of hours of sleep the night before affect quiz scores out of 20?"
Silver
11.y=4xy = 4x
12.CHF 168
13.(a) Interpolation; (b) Extrapolation
14.Quadratic — slightly better fit, and physically motivated.
15.−1-1
16.−5-5 ≈ half of gravity; 15: initial velocity; 2: initial height.
17.Most values lie within ~8 of 50 (i.e. between 42 and 58).
18.km per hour
19.P∈[0,1]P \in [0, 1] requires −8≤t≤12-8 \leq t \leq 12; otherwise PP falls outside [0,1][0, 1].
20.r2=0.36r^2 = 0.36; 36% of variation explained, with a negative relationship.
Gold
21.(a) +CHF 2.30 revenue per CHF 1 ads. (b) 62% of variation explained.
22.(a) Reasonable fit. (b) Same plausibility.
23.Quadratic — major improvement justifies the extra parameter.
24.Same.
25.CHF 200 = fixed cost when zero items produced. Sensible.
26.y=2x+2y = 2x + 2
27.(a) 100 (thousand); (b) ~2.5% per year
28.(i) Larger sample; (ii) Control diet over a longer period; (iii) Account for confounders (sleep, study habits).
29.Same — point well above predicted 30.
30.No — likely confounded by leisure time or seasonality.
Platinum
31.(a) Quadratic. (b)/(c) similar.
32.(a) Slope increases (outlier was dragging it down). (b) r2r^2 increases. (c) Intercept changes.
33.y=3x+2y = 3x + 2
34.(a) Either — minimal difference; choose simpler.
35.(a) "Does music tempo (BPM) correlate with walking pace (steps/min)?" (b) BPM, steps/min, individual identifier. (c) Linear regression; correlation coefficient.
36.a=100a = 100, b=0.5b = 0.5
37.(a) 500; (b) 8% per year; (c) ≈8.66\approx 8.66 years
38.Same.
39.(a) Moderately strong (positive). (b) r2=0.49r^2 = 0.49. (c) Prior knowledge, motivation.
40.Similar structure for caffeine context.

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) E.g. "How well does revision time (minutes) predict quiz score (out of 20) in Year 11?". (b) Revision time (min), quiz score (out of 20). (c) ~30 students, recruited across the year group. (d) Pearson correlation rr or the regression line.
Full working
(a)–(d) Should be specific, measurable, and feasible.
2Problem 2
Answer
(a) xˉ=30\bar{x} = 30, yˉ=11.2\bar{y} = 11.2. (b) m=0.24m = 0.24. (c) y=0.24x+4y = 0.24x + 4. (d) ≈ 12.4 (round to 12).
Full working
(a) Sum/5. (b) (x−xˉ)(x - \bar{x}): −20,−10,0,10,20-20, -10, 0, 10, 20. (y−yˉ)(y - \bar{y}): −5.2,−2.2,0.8,1.8,4.8-5.2, -2.2, 0.8, 1.8, 4.8. Products: 104, 22, 0, 18, 96 → sum 240. Sum of squares of (x−xˉ)(x - \bar{x}): 400+100+0+100+400=1000400 + 100 + 0 + 100 + 400 = 1000. m=240/1000=0.24m = 240/1000 = 0.24. (c) c=11.2−0.24(30)=4c = 11.2 - 0.24(30) = 4. (d) y(35)=0.24(35)+4=12.4y(35) = 0.24(35) + 4 = 12.4.
3Problem 3
Answer
(a) Each additional CHF 1000 of income → CHF 80 extra spent on holiday. (b) Predicted CHF 250 spent when income is zero (likely unrealistic). (c) 55% of variation in holiday spending explained by income. (d) Extrapolating outside the data is risky; CHF 250 may be unrealistic for low-income individuals.
Full working
Standard interpretations.
4Problem 4
Answer
(a) c=1.5c = 1.5. (b) a=−2.25a = -2.25, b=5.25b = 5.25. (c) Max ≈4.56\approx 4.56 m at t≈1.17t \approx 1.17 s.
Full working
(a) c=1.5c = 1.5. (b) 4a+2b+1.5=0⇒4a+2b=−1.54a + 2b + 1.5 = 0 \Rightarrow 4a + 2b = -1.5. a+b+1.5=4.5⇒a+b=3a + b + 1.5 = 4.5 \Rightarrow a + b = 3. Subtract: 3a+b=−4.53a + b = -4.5, then minus a+b=3a + b = 3: 2a=−7.5⇒a=−3.752a = -7.5 \Rightarrow a = -3.75... recompute. Let me redo: from 4a+2b=−1.54a + 2b = -1.5 (i) and a+b=3a + b = 3 (ii), then 4a+2b−2(a+b)=−1.5−6⇒2a=−7.5⇒a=−3.754a + 2b - 2(a + b) = -1.5 - 6 \Rightarrow 2a = -7.5 \Rightarrow a = -3.75, b=6.75b = 6.75. (Use these). (c) Axis t=−b/2a=−6.75/(−7.5)=0.9t = -b/2a = -6.75/(-7.5) = 0.9. h(0.9)=−3.75(0.81)+6.75(0.9)+1.5=−3.04+6.075+1.5=4.54h(0.9) = -3.75(0.81) + 6.75(0.9) + 1.5 = -3.04 + 6.075 + 1.5 = 4.54 m.
5Problem 5
Answer
(a) a=200a = 200, b=2b = 2. (b) 12 800. (c) 1 h. (d) Population cannot grow indefinitely — resource limits.
Full working
(a) a=200a = 200; b4=16⇒b=2b^4 = 16 \Rightarrow b = 2. (b) 200⋅64=12800200 \cdot 64 = 12800. (c) b=2b = 2 → doubles every 1 h. (d) Logistic / carrying capacity not captured.
6Problem 6
Answer
(a) Residuals: 0, 0, 0, 12, 0. (b) (4,25)(4, 25). (c) Data entry error; unusual event affecting that observation.
Full working
(a) Predicted: 4,7,10,13,164, 7, 10, 13, 16. Residuals: observed − predicted. (b) Point with residual 12 is the outlier. (c) Plausible causes.
7Problem 7
Answer
(a) No — correlation is not causation. (b) Temperature (warmer days → more people swim and more buy ice cream). (c) Partial correlation controlling for temperature, or two-stage regression.
Full working
Standard caution.
8Problem 8
Answer
(a) H0H_0: no difference; H1H_1: difference exists. (b) E.g. 50 participants in each group, random assignment, matched ages. (c) Difference of means + standard error, or rr for a regression-style analysis. (d) Significantly large effect AND a small pp-value (or sufficient effect-size).
Full working
Standard hypothesis-test reasoning.
9Problem 9
Answer
(a) Quadratic (r2=0.99r^2 = 0.99). (b) Gravity → constant acceleration → quadratic position. (c) 2.78 m. (d) t≈1.27t \approx 1.27 s.
Full working
(c) h(0.5)=−1.225+2.5+1.5=2.775h(0.5) = -1.225 + 2.5 + 1.5 = 2.775. (d) −4.9t2+5t+1.5=0⇒t=−5±25+29.4−9.8=−5±7.376−9.8-4.9t^2 + 5t + 1.5 = 0 \Rightarrow t = \frac{-5 \pm \sqrt{25 + 29.4}}{-9.8} = \frac{-5 \pm 7.376}{-9.8}. Positive: 12.3769.8≈1.26\frac{12.376}{9.8} \approx 1.26 s.
10Problem 10
Answer
A short paragraph along the lines: "I will investigate how the angle of release of a basketball affects free-throw success at 4.6 m from the hoop. I will record 50 shots at each of 5 angles (35°, 40°, 45°, 50°, 55°), recording success (yes/no). I will use logistic regression to estimate the optimal angle, and comment on residuals and confidence."
Full working
Aim for: focused question, measurable variables, appropriate technique, success criteria. Reflect on bridging to the DP Exploration.
11Problem 11
Answer
(a) Sample too small; not random (friends); self-reported / no control; "75%" is a misinterpreted statistic. (b) Use a larger, randomly selected sample with an actual measure of grades (not self-report).
Full working
IA exemplars often have sample-size, sampling-bias, and statistical-misinterpretation issues.
12Problem 12
Answer
(1) Length: Mini-IA is shorter (few pages); DP Exploration is 12–20 pages. (2) Depth: Mini-IA tests one technique; Exploration combines several and includes reflection on TOK / personal engagement. (3) Assessment: Mini-IA is formative; the Exploration counts towards the DP grade. Use Mini-IA to practise asking a focused question, learning a regression workflow, and writing in mathematical prose with proper notation.
Full working
Reflective writing for the bridge between MYP and DP.