Problems — 11.12 Mini IA — DP exploration practice

1

**Modelling cycle — basic.** A student wants to explore "does revision time predict quiz score?". (a) State a specific research question with units. (b) List variables you would collect (with units). (c) Suggest a sample size and how to recruit participants. (d) Suggest a primary statistic to report.

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2

**Linear fit by hand.** A student collected: | Revision (min) | 10 | 20 | 30 | 40 | 50 | |---|---|---|---|---|---| | Score | 6 | 9 | 12 | 13 | 16 | (a) Find the means $\bar{x}$ and $\bar{y}$. (b) Compute the gradient using $m = \dfrac{\sum(x - \bar{x})(y - \bar{y})}{\sum(x - \bar{x})^2}$. (c) Write the regression line $y = mx + c$. (d) Predict the score for 35 min of revision.

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3

**Interpreting regression output.** A regression of $y$ (CHF spent on holiday) against $x$ (annual income in 1000s CHF) gives $y = 0.08x + 250$, $r^2 = 0.55$. (a) State the meaning of the gradient. (b) State the meaning of the intercept. (c) State the meaning of $r^2 = 0.55$. (d) Critique using the model for $x = 0$ (someone with no income).

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4

**Quadratic model fit.** A student records the height of a ball thrown vertically: | $t$ (s) | 0 | 0.5 | 1.0 | 1.5 | 2.0 | |---|---|---|---|---|---| | $h$ (m) | 1.5 | 4.0 | 4.5 | 3.0 | 0 | (a) Assume $h(t) = at^2 + bt + c$. From $h(0)$, find $c$. (b) Use $h(2) = 0$ and $h(1) = 4.5$ to set up two equations and solve for $a$ and $b$. (c) State the maximum height and when it occurs.

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5

**Exponential fit by two points.** A bacteria colony's data: | $t$ (h) | 0 | 4 | |---|---|---| | $N$ | 200 | 3200 | (a) Fit a model $N(t) = a \cdot b^t$. (b) Predict the population at $t = 6$ h. (c) Find the doubling time, to 3 s.f. (d) State one limitation of using this model for very long times.

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6

**Residuals and outlier detection.** A regression $y = 3x + 1$ fits well except for one point. Observed data: | $x$ | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | $y$ | 4 | 7 | 10 | 25 | 16 | (a) Compute the residuals. (b) Identify the outlier. (c) Suggest two possible explanations.

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7

**Correlation vs causation.** A study finds $r = 0.85$ between ice-cream sales and shark attacks across summer months. (a) Does this mean ice cream causes shark attacks? Justify. (b) Identify a confounding variable. (c) Suggest one further analysis that would help clarify the relationship.

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8

**Investigating a hypothesis end-to-end.** A student investigates whether wearing glasses affects reaction-time score on an online test. (a) State a null hypothesis and an alternative hypothesis. (b) Design a sample plan: target sample size, recruitment, controls. (c) Suggest a statistic to test the hypothesis. (d) Comment on what would constitute "evidence to reject the null".

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9

**Quadratic vs linear modelling.** A student fits two models to the same data: - Linear: $h = -3.2t + 2.0$, $r^2 = 0.20$ - Quadratic: $h = -4.9t^2 + 5.0t + 1.5$, $r^2 = 0.99$ (a) Which fits better, and why? (b) State one physical reason the quadratic is more appropriate. (c) Predict the height at $t = 0.5$ s using Model 2. (d) Predict when the ball returns to the floor (use the quadratic formula).

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10

**Mini-IA proposal.** Write a one-paragraph proposal for a Mini-IA project. Your proposal should include: (a) A specific research question. (b) The variables you will collect (with units). (c) The mathematical technique you plan to use. (d) How you will judge whether your analysis answers the question. (Sample answer: an exemplar paragraph.)

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11

**Critique of an IA exemplar.** An IA report states: "I asked 8 friends if eating breakfast affected their grades. I found that 6 of 8 reported higher grades, so eating breakfast improves grades by 75%." (a) Identify three methodological weaknesses. (b) Suggest a single concrete improvement that would address the biggest weakness.

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12

**Bridge to the DP Exploration.** Describe **three** ways a Year-11 Mini-IA differs from a full DP Year-12 Exploration, and suggest how to use the Mini-IA as preparation for the larger project.

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