Solutions — Full Answer Key

Mathematics

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:

T

; Independent:

h

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 - 1

12.CHF 170

13.(a) Interpolation; (b) Extrapolation

14.Quadratic, because

r^2

is much higher and gravity gives quadratic motion.

15.1

16.

-4.9

: half of gravitational acceleration (=

-\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 = 0

(where

P = 1

, the maximum). For

t > 0

the model gives

P > 1

, which is impossible.

20.

r^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

r^2

. (Occam.)

24.(1) There is a theoretical reason to expect the model holds beyond observed range; (2) Sensible behaviour at

x = 50

(no breakdown).

25.CHF 250 = expected revenue with zero advertising spend (baseline). Plausible if there's organic demand.

26.

y = 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

y

-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 \approx 1.27

32.(a) Slope decreases (less pulled up by outlier). (b)

r^2

increases (less noise). (c) Intercept rises slightly.

33.

y = 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 = 5

b = 2

37.(a) 1000; (b) 5% per year; (c)

\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)

r^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:

C

; Independent:

n

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 = 4x

12.CHF 168

13.(a) Interpolation; (b) Extrapolation

14.Quadratic — slightly better fit, and physically motivated.

15.

-1

16.

-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 \in [0, 1]

requires

-8 \leq t \leq 12

; otherwise

P

falls outside

[0, 1]

20.

r^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 + 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)

r^2

increases. (c) Intercept changes.

33.

y = 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 = 100

b = 0.5

37.(a) 500; (b) 8% per year; (c)

\approx 8.66

years

38.Same.

39.(a) Moderately strong (positive). (b)

r^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

r

or the regression line.

Full working

(a)–(d) Should be specific, measurable, and feasible.

2Problem 2

Answer

(a)

\bar{x} = 30

\bar{y} = 11.2

. (b)

m = 0.24

. (c)

y = 0.24x + 4

. (d) ≈ 12.4 (round to 12).

Full working

(a) Sum/5. (b)

(x - \bar{x})

-20, -10, 0, 10, 20

(y - \bar{y})

-5.2, -2.2, 0.8, 1.8, 4.8

. Products: 104, 22, 0, 18, 96 → sum 240. Sum of squares of

(x - \bar{x})

400 + 100 + 0 + 100 + 400 = 1000

m = 240/1000 = 0.24

. (c)

c = 11.2 - 0.24(30) = 4

. (d)

y(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 = 1.5

. (b)

a = -2.25

b = 5.25

. (c) Max

\approx 4.56

m at

t \approx 1.17

Full working

(a)

c = 1.5

. (b)

4a + 2b + 1.5 = 0 \Rightarrow 4a + 2b = -1.5

a + b + 1.5 = 4.5 \Rightarrow a + b = 3

. Subtract:

3a + b = -4.5

, then minus

a + b = 3

2a = -7.5 \Rightarrow a = -3.75

... recompute. Let me redo: from

4a + 2b = -1.5

(i) and

a + b = 3

(ii), then

4a + 2b - 2(a + b) = -1.5 - 6 \Rightarrow 2a = -7.5 \Rightarrow a = -3.75

b = 6.75

. (Use these). (c) Axis

t = -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 = 4.54

5Problem 5

Answer

(a)

a = 200

b = 2

. (b) 12 800. (c) 1 h. (d) Population cannot grow indefinitely — resource limits.

Full working

(a)

a = 200

;

b^4 = 16 \Rightarrow b = 2

. (b)

200 \cdot 64 = 12800

. (c)

b = 2

→ doubles every 1 h. (d) Logistic / carrying capacity not captured.

6Problem 6

Answer

(a) Residuals: 0, 0, 0, 12, 0. (b)

(4, 25)

. (c) Data entry error; unusual event affecting that observation.

Full working

(a) Predicted:

4, 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)

H_0

: no difference;

H_1

: difference exists. (b) E.g. 50 participants in each group, random assignment, matched ages. (c) Difference of means + standard error, or

r

for a regression-style analysis. (d) Significantly large effect AND a small

p

-value (or sufficient effect-size).

Full working

Standard hypothesis-test reasoning.

9Problem 9

Answer

(a) Quadratic (

r^2 = 0.99

). (b) Gravity → constant acceleration → quadratic position. (c) 2.78 m. (d)

t \approx 1.27

Full working

(c)

h(0.5) = -1.225 + 2.5 + 1.5 = 2.775

. (d)

-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:

\frac{12.376}{9.8} \approx 1.26

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.