Topic 4.13: Non-linear regression (HL only) Questions

Using a model fitted to ages 5–15 to predict a value at age 60 is an example of:

Data that increases by a constant FACTOR each step is best modelled by:

A GDC reports R² = 0.999 for a regression model. This means:

Find the predicted value of $y$ when $x=8$.

Bacteria in a culture are counted each hour: t = 0,1,2,3,4 hours give N = 50, 78, 122, 190, 297 (cells). Fit a model N = k·aᵗ and state R².

The sum of squared residuals SSres for a model is:

Two models fit the same data: a quadratic (R² = 0.95) and a cubic (R² = 0.951). The better practical choice is usually:

A model predicts heights ŷ = 3.1, 6.0, 9.2, 11.8 for data y = 3.0, 6.4, 9.0, 12.0. Find the sum of squared residuals SSres.

Daily high temperature T (°C) through a year repeats with the seasons. A sinusoidal model T = 9·sin(0.017·d − 1.4) + 15 is fitted, where d is the day number (1–365), with R² = 0.97. State the highest temperature the model predicts and comment on the model's suitability.

The brightness B (lux) of a lamp is measured at distance d (m): d = 1,2,3,4 give B = 100, 25, 11, 6.3. Fit a power model B = a·dᵇ, state R², and interpret the value of b.

A profit P (thousand $) over a launch is modelled by a quadratic P = −0.5n² + 6n − 4, fitted to monthly data with R² = 0.91. (a) Estimate the profit in month n = 5. (b) The same data fitted with a cubic gives R² = 0.93. State which model you would report and why.

A radioactive sample's mass M (grams) is recorded: t = 0,2,4,6 days give M = 80, 50, 31, 20. (a) Fit M = k·eʳᵗ and state r. (b) Use the model to estimate when the mass falls to 10 g.