Answer Key
11.12 Mini IA — DP exploration practice
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | In the model $T = 0.05h + 0.55$ (with $T$ in seconds, $h$ in hours of sleep), name the dependent and independent variables. | Dependent: $T$; Independent: $h$. |
| 2 | A linear model $C = 3n + 5$ gives the cost (CHF) of $n$ items. State what $m$ and $c$ represent. | CHF 3 per item; fixed cost CHF 5 |
| 3 | A linear regression has $r^2 = 0.85$. State in words what this means. | 85% of the variation in the dependent variable is explained by the model. |
| 4 | Find the mean of $\{4, 7, 9, 11, 14\}$. | 9 |
| 5 | Using $C = 0.5n + 20$, predict the cost for $n = 100$. | 70 |
| 6 | A scatter plot shows ice cream sales rising as temperature rises. State the kind of correlation. | Positive (direct) correlation. |
| 7 | A graph of a tossed-ball height vs time looks like a downward parabola. What kind of model is appropriate? | Quadratic. |
| 8 | In a regression, the **residual** at a data point is defined as which of the following? | (i) observed − predicted |
| 9 | A scatter plot shows points roughly forming an "S"-curve, levelling at the top. Suggest a model type. | Logistic model (or saturating curve). |
| 10 | An IA mini-project must have a focused research question. Suggest a research question for an investigation of caffeine intake and reaction time. | "How does caffeine intake (mg) affect reaction time (ms) in young adults?" |
| 11 | Find the linear model through $(2, 5)$ and $(6, 17)$. | $y = 3x - 1$ |
| 12 | For the ice-cream sales model $y = 6x + 20$ (sales CHF, $x$ = temperature °C), predict sales at 25°C. | CHF 170 |
| 13 | A dataset covers ages 12–18. Distinguish whether predicting at age (a) 14, (b) 25 is interpolation or extrapolation. | (a) Interpolation; (b) Extrapolation |
| 14 | A model is fitted to height vs time of a tossed ball. Two models give: linear $h = -3.2t + 2$ ($r^2 = 0.20$) and quadratic $h = -4.9t^2 + 5t + 1.5$ ($r^2 = 0.99$). Which fits better, and why? | Quadratic, because $r^2$ is much higher and gravity gives quadratic motion. |
| 15 | The line of best fit is $y = 2x + 1$. The observed value at $x = 3$ is 8. Find the residual. | 1 |
| 16 | For the projectile model $h(t) = -4.9t^2 + 20t + 1$, state the meaning of the values $-4.9$, $20$, and $1$. | $-4.9$: half of gravitational acceleration (= $-\tfrac{1}{2}g$); 20: initial velocity (m/s); 1: initial height (m). |
| 17 | A small dataset has mean 12 and standard deviation 3. Suggest a one-sentence summary about the data's spread. | Most values lie within ~3 of 12 (i.e. between 9 and 15). |
| 18 | A model $C = 0.5n + 20$ gives cost in CHF for $n$ items. State the units of the gradient. | CHF per item |
| 19 | A model $P = 0.2t + 1$ gives a person's probability of catching a cold against time. For what range of $t$ does the model make sense? | Only for $t = 0$ (where $P = 1$, the maximum). For $t > 0$ the model gives $P > 1$, which is impossible. |
| 20 | A regression report gives $r = 0.80$. Find $r^2$ and interpret. | $r^2 = 0.64$; 64% of variation explained. |
| 21 | A linear regression on weekly revenue (CHF) vs advertising spend (CHF) gives $R = 1.8A + 250$ with $r^2 = 0.84$. State (a) the practical meaning of the gradient and (b) of $r^2$. | (a) Each CHF 1 of advertising → CHF 1.80 extra revenue. (b) 84% of variation in revenue explained by advertising. |
| 22 | A student fits $T = -0.05h + 0.55$ for reaction time vs sleep hours, with $r^2 = 0.38$. Comment on whether (a) the model fits well and (b) the sign of the gradient is plausible. | (a) Weak fit (only 38% explained); (b) Negative gradient means more sleep → faster reaction; plausible direction. |
| 23 | Two models fit a dataset: linear $r^2 = 0.95$ and quadratic $r^2 = 0.96$. Which is preferable, and why? | Linear is preferable — extra parameter of quadratic barely improves $r^2$. (Occam.) |
| 24 | A model fits data from $x = 1$ to $x = 10$. Using the model to predict $y$ at $x = 50$ is reasonable only when (state two conditions). | (1) There is a theoretical reason to expect the model holds beyond observed range; (2) Sensible behaviour at $x = 50$ (no breakdown). |
| 25 | In the model $R = 1.8A + 250$ from G1, interpret the value 250 in context. Is it physically meaningful? | CHF 250 = expected revenue with zero advertising spend (baseline). Plausible if there's organic demand. |
| 26 | Given two data points $(0, 5)$ and $(10, 35)$, find a linear model $y = mx + c$. | $y = 3x + 5$ |
| 27 | A model fits the population (thousands) of a town: $P(t) = 50 e^{0.04t}$ where $t$ is years from 2000. State (a) the population in 2000, (b) the annual growth rate. | (a) 50 (thousand); (b) ~4% per year |
| 28 | A student is investigating "does music affect concentration?". Suggest **three** improvements to a sample of 10 students filling out a 5-question quiz once. | (i) Larger sample; (ii) Random assignment of music vs no-music; (iii) Repeated trials. |
| 29 | A regression gives $y = 2x + 3$, $r^2 = 0.92$. One data point $(8, 50)$ has a residual of 31. Comment. | The point is an outlier — its $y$-value is far above the predicted 19. Investigate the cause (data entry error? unusual circumstance?). |
| 30 | A study finds $r = 0.85$ between ice-cream sales and shark attacks across summer months. Does this mean ice-cream sales cause shark attacks? | No — correlation does not imply causation. Both are driven by temperature (a confounding variable). |
| 31 | A student fits two models to height-vs-time data of a tossed ball: linear $h = -3.2t + 2$ with $r^2 = 0.20$, and quadratic $h = -4.9t^2 + 5t + 1.5$ with $r^2 = 0.99$. (a) Which fits better and why? (b) Predict the height at $t = 0.5$ s using the better model. (c) Predict the time when the ball hits the floor. | (a) Quadratic. (b) 2.78 m. (c) $t \approx 1.27$ s. |
| 32 | A small dataset has 9 points with a strong linear pattern and an outlier $(20, 100)$ when the rest cluster between $(0, 0)$ and $(10, 50)$. Predict qualitatively the effect of removing the outlier on (a) the slope of the line of best fit, (b) $r^2$, (c) the intercept. | (a) Slope decreases (less pulled up by outlier). (b) $r^2$ increases (less noise). (c) Intercept rises slightly. |
| 33 | Three points $(1, 4)$, $(2, 7)$, $(3, 8)$ are given. Find the equation of the line of best fit using the formula $m = \frac{\sum(x - \bar{x})(y - \bar{y})}{\sum(x - \bar{x})^2}$. | $y = 2x + 2.33$ approx |
| 34 | For an investigation on caffeine and reaction time, Model A: $T = -0.005c + 0.45$, $r^2 = 0.42$; Model B: $T = 0.001 c^2 - 0.012 c + 0.45$, $r^2 = 0.78$. (a) Which model fits better? (b) State a physical reason a non-monotonic model might be plausible. (c) State a limitation of using the better model to make predictions. | (a) Model B. (b) Caffeine improves reactions up to some optimum, then worsens (jittery). (c) Extrapolation beyond data range is risky. |
| 35 | A student wants to investigate basketball free-throw success. Suggest (a) a specific research question, (b) the variables to collect, (c) two mathematical techniques you would use. | (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 | Two data points lie on an exponential model $y = a \cdot b^x$: $(0, 5)$ and $(3, 40)$. Find $a$ and $b$. | $a = 5$, $b = 2$ |
| 37 | A growth model is $P(t) = 1000 e^{0.05t}$. State (a) the initial value, (b) the continuous growth rate as a percentage, (c) the doubling time (to 3 s.f.). | (a) 1000; (b) 5% per year; (c) $\approx 13.9$ years |
| 38 | A height-vs-age model $H(a) = 70 + 5a$ (cm, $a$ years) is fitted to children aged 2–10. State two problems with using the model for (a) babies (age 0–1), (b) teenagers (age 14+). | (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 correlation $r = -0.4$ is observed between hours of TV and hours of sleep. (a) Is the relationship strong, moderate, or weak? (b) What is $r^2$? (c) State one possible confounder. | (a) Moderate (negative). (b) $r^2 = 0.16$ → 16% of variation explained. (c) Age, screen-time policy, household routine. |
| 40 | For an investigation: (a) state a testable hypothesis about sleep and quiz score; (b) propose a 5-step data collection plan; (c) write a sentence about what would convince you to reject the hypothesis. | (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
| # | Question | Answer |
|---|---|---|
| 1 | In the model $T = 0.05h + 0.55$ (with $T$ in seconds, $h$ in hours of sleep), name the dependent and independent variables. | Dependent: $C$; Independent: $n$. |
| 2 | A linear model $C = 4n + 2$ gives the cost (CHF) of $n$ items. State what $m$ and $c$ represent. | CHF 4 per item; fixed cost CHF 2 |
| 3 | A linear regression has $r^2 = 0.85$. State in words what this means. | 30% of variation is explained — fit is weak. |
| 4 | Find the mean of $\{4, 7, 9, 11, 14\}$. | 7 |
| 5 | Using $C = 0.5n + 20$, predict the cost for $n = 100$. | 11 |
| 6 | A scatter plot shows ice cream sales rising as temperature rises. State the kind of correlation. | Negative (inverse) correlation. |
| 7 | A graph of a tossed-ball height vs time looks like a downward parabola. What kind of model is appropriate? | Exponential. |
| 8 | In a regression, the **residual** at a data point is defined as which of the following? | (i) observed − predicted |
| 9 | A scatter plot shows points roughly forming an "S"-curve, levelling at the top. Suggest a model type. | Linear. |
| 10 | An IA mini-project must have a focused research question. Suggest a research question for an investigation of caffeine intake and reaction time. | "How does the number of hours of sleep the night before affect quiz scores out of 20?" |
| 11 | Find the linear model through $(2, 5)$ and $(6, 17)$. | $y = 4x$ |
| 12 | For the ice-cream sales model $y = 6x + 20$ (sales CHF, $x$ = temperature °C), predict sales at 25°C. | CHF 168 |
| 13 | A dataset covers ages 12–18. Distinguish whether predicting at age (a) 14, (b) 25 is interpolation or extrapolation. | (a) Interpolation; (b) Extrapolation |
| 14 | A model is fitted to height vs time of a tossed ball. Two models give: linear $h = -3.2t + 2$ ($r^2 = 0.20$) and quadratic $h = -4.9t^2 + 5t + 1.5$ ($r^2 = 0.99$). Which fits better, and why? | Quadratic — slightly better fit, and physically motivated. |
| 15 | The line of best fit is $y = 2x + 1$. The observed value at $x = 3$ is 8. Find the residual. | $-1$ |
| 16 | For the projectile model $h(t) = -4.9t^2 + 20t + 1$, state the meaning of the values $-4.9$, $20$, and $1$. | $-5$ ≈ half of gravity; 15: initial velocity; 2: initial height. |
| 17 | A small dataset has mean 12 and standard deviation 3. Suggest a one-sentence summary about the data's spread. | Most values lie within ~8 of 50 (i.e. between 42 and 58). |
| 18 | A model $C = 0.5n + 20$ gives cost in CHF for $n$ items. State the units of the gradient. | km per hour |
| 19 | A model $P = 0.2t + 1$ gives a person's probability of catching a cold against time. For what range of $t$ does the model make sense? | $P \in [0, 1]$ requires $-8 \leq t \leq 12$; otherwise $P$ falls outside $[0, 1]$. |
| 20 | A regression report gives $r = 0.80$. Find $r^2$ and interpret. | $r^2 = 0.36$; 36% of variation explained, with a negative relationship. |
| 21 | A linear regression on weekly revenue (CHF) vs advertising spend (CHF) gives $R = 1.8A + 250$ with $r^2 = 0.84$. State (a) the practical meaning of the gradient and (b) of $r^2$. | (a) +CHF 2.30 revenue per CHF 1 ads. (b) 62% of variation explained. |
| 22 | A student fits $T = -0.05h + 0.55$ for reaction time vs sleep hours, with $r^2 = 0.38$. Comment on whether (a) the model fits well and (b) the sign of the gradient is plausible. | (a) Reasonable fit. (b) Same plausibility. |
| 23 | Two models fit a dataset: linear $r^2 = 0.95$ and quadratic $r^2 = 0.96$. Which is preferable, and why? | Quadratic — major improvement justifies the extra parameter. |
| 24 | A model fits data from $x = 1$ to $x = 10$. Using the model to predict $y$ at $x = 50$ is reasonable only when (state two conditions). | Same. |
| 25 | In the model $R = 1.8A + 250$ from G1, interpret the value 250 in context. Is it physically meaningful? | CHF 200 = fixed cost when zero items produced. Sensible. |
| 26 | Given two data points $(0, 5)$ and $(10, 35)$, find a linear model $y = mx + c$. | $y = 2x + 2$ |
| 27 | A model fits the population (thousands) of a town: $P(t) = 50 e^{0.04t}$ where $t$ is years from 2000. State (a) the population in 2000, (b) the annual growth rate. | (a) 100 (thousand); (b) ~2.5% per year |
| 28 | A student is investigating "does music affect concentration?". Suggest **three** improvements to a sample of 10 students filling out a 5-question quiz once. | (i) Larger sample; (ii) Control diet over a longer period; (iii) Account for confounders (sleep, study habits). |
| 29 | A regression gives $y = 2x + 3$, $r^2 = 0.92$. One data point $(8, 50)$ has a residual of 31. Comment. | Same — point well above predicted 30. |
| 30 | A study finds $r = 0.85$ between ice-cream sales and shark attacks across summer months. Does this mean ice-cream sales cause shark attacks? | No — likely confounded by leisure time or seasonality. |
| 31 | A student fits two models to height-vs-time data of a tossed ball: linear $h = -3.2t + 2$ with $r^2 = 0.20$, and quadratic $h = -4.9t^2 + 5t + 1.5$ with $r^2 = 0.99$. (a) Which fits better and why? (b) Predict the height at $t = 0.5$ s using the better model. (c) Predict the time when the ball hits the floor. | (a) Quadratic. (b)/(c) similar. |
| 32 | A small dataset has 9 points with a strong linear pattern and an outlier $(20, 100)$ when the rest cluster between $(0, 0)$ and $(10, 50)$. Predict qualitatively the effect of removing the outlier on (a) the slope of the line of best fit, (b) $r^2$, (c) the intercept. | (a) Slope increases (outlier was dragging it down). (b) $r^2$ increases. (c) Intercept changes. |
| 33 | Three points $(1, 4)$, $(2, 7)$, $(3, 8)$ are given. Find the equation of the line of best fit using the formula $m = \frac{\sum(x - \bar{x})(y - \bar{y})}{\sum(x - \bar{x})^2}$. | $y = 3x + 2$ |
| 34 | For an investigation on caffeine and reaction time, Model A: $T = -0.005c + 0.45$, $r^2 = 0.42$; Model B: $T = 0.001 c^2 - 0.012 c + 0.45$, $r^2 = 0.78$. (a) Which model fits better? (b) State a physical reason a non-monotonic model might be plausible. (c) State a limitation of using the better model to make predictions. | (a) Either — minimal difference; choose simpler. |
| 35 | A student wants to investigate basketball free-throw success. Suggest (a) a specific research question, (b) the variables to collect, (c) two mathematical techniques you would use. | (a) "Does music tempo (BPM) correlate with walking pace (steps/min)?" (b) BPM, steps/min, individual identifier. (c) Linear regression; correlation coefficient. |
| 36 | Two data points lie on an exponential model $y = a \cdot b^x$: $(0, 5)$ and $(3, 40)$. Find $a$ and $b$. | $a = 100$, $b = 0.5$ |
| 37 | A growth model is $P(t) = 1000 e^{0.05t}$. State (a) the initial value, (b) the continuous growth rate as a percentage, (c) the doubling time (to 3 s.f.). | (a) 500; (b) 8% per year; (c) $\approx 8.66$ years |
| 38 | A height-vs-age model $H(a) = 70 + 5a$ (cm, $a$ years) is fitted to children aged 2–10. State two problems with using the model for (a) babies (age 0–1), (b) teenagers (age 14+). | Same. |
| 39 | A correlation $r = -0.4$ is observed between hours of TV and hours of sleep. (a) Is the relationship strong, moderate, or weak? (b) What is $r^2$? (c) State one possible confounder. | (a) Moderately strong (positive). (b) $r^2 = 0.49$. (c) Prior knowledge, motivation. |
| 40 | For an investigation: (a) state a testable hypothesis about sleep and quiz score; (b) propose a 5-step data collection plan; (c) write a sentence about what would convince you to reject the hypothesis. | Similar structure for caffeine context. |
Problems — Worked Solutions
**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.
(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.
**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.
(a) $\bar{x} = 30$, $\bar{y} = 11.2$. (b) $m = 0.24$. (c) $y = 0.24x + 4$. (d) ≈ 12.4 (round to 12).
**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).
(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.
**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.
(a) $c = 1.5$. (b) $a = -2.25$, $b = 5.25$. (c) Max $\approx 4.56$ m at $t \approx 1.17$ s.
**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.
(a) $a = 200$, $b = 2$. (b) 12 800. (c) 1 h. (d) Population cannot grow indefinitely — resource limits.
**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.
(a) Residuals: 0, 0, 0, 12, 0. (b) $(4, 25)$. (c) Data entry error; unusual event affecting that observation.
**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.
(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.
**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".
(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).
**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).
(a) Quadratic ($r^2 = 0.99$). (b) Gravity → constant acceleration → quadratic position. (c) 2.78 m. (d) $t \approx 1.27$ s.
**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.)
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."
**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.
(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).
**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.
(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.