11.12 Mini IA — DP exploration practice

In the model $T = 0.05h + 0.55$ (with $T$ in seconds, $h$ in hours of sleep), name the dependent and independent variables.

A linear model $C = 3n + 5$ gives the cost (CHF) of $n$ items. State what $m$ and $c$ represent.

A linear regression has $r^2 = 0.85$. State in words what this means.

Find the mean of $\{4, 7, 9, 11, 14\}$.

Using $C = 0.5n + 20$, predict the cost for $n = 100$.

A scatter plot shows ice cream sales rising as temperature rises. State the kind of correlation.

A graph of a tossed-ball height vs time looks like a downward parabola. What kind of model is appropriate?

In a regression, the **residual** at a data point is defined as which of the following?

A scatter plot shows points roughly forming an "S"-curve, levelling at the top. Suggest a model type.

An IA mini-project must have a focused research question. Suggest a research question for an investigation of caffeine intake and reaction time.

Find the linear model through $(2, 5)$ and $(6, 17)$.

For the ice-cream sales model $y = 6x + 20$ (sales CHF, $x$ = temperature °C), predict sales at 25°C.

A dataset covers ages 12–18. Distinguish whether predicting at age (a) 14, (b) 25 is interpolation or extrapolation.

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?

The line of best fit is $y = 2x + 1$. The observed value at $x = 3$ is 8. Find the residual.

For the projectile model $h(t) = -4.9t^2 + 20t + 1$, state the meaning of the values $-4.9$, $20$, and $1$.

A small dataset has mean 12 and standard deviation 3. Suggest a one-sentence summary about the data's spread.

A model $C = 0.5n + 20$ gives cost in CHF for $n$ items. State the units of the gradient.

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?

A regression report gives $r = 0.80$. Find $r^2$ and interpret.

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 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.

Two models fit a dataset: linear $r^2 = 0.95$ and quadratic $r^2 = 0.96$. Which is preferable, and why?

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).

In the model $R = 1.8A + 250$ from G1, interpret the value 250 in context. Is it physically meaningful?

Given two data points $(0, 5)$ and $(10, 35)$, find a linear model $y = mx + c$.

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 student is investigating "does music affect concentration?". Suggest **three** improvements to a sample of 10 students filling out a 5-question quiz once.

A regression gives $y = 2x + 3$, $r^2 = 0.92$. One data point $(8, 50)$ has a residual of 31. Comment.

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?

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 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.

Three points $(1, 4)$, $(2, 7)$, $(3, 8)$ are given. Find the equation of the line of best fit using the formula $m = \frac{\sum(x - \bar{x})(y - \bar{y})}{\sum(x - \bar{x})^2}$.

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 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.

Two data points lie on an exponential model $y = a \cdot b^x$: $(0, 5)$ and $(3, 40)$. Find $a$ and $b$.

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 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 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.

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.