AHL 4.13—Non-linear regression Questionbank

A biologist is studying the growth of a bacterial colony. The number of bacteria, $B$ (in thousands), after $t$ hours is recorded as follows:

The biologist models the growth using a logistic function of the form $B=\frac{L}{1+C e^{-k t}}$, where $L, C$, and $k$ are positive constants, and $L$ represents the carrying capacity.

Elena is modeling the decay of a radioactive substance. The mass, $M$ (in grams), remaining after $t$ days is given by:

She proposes a model of the form $M=M_{0} e^{-k t}$, where $M_{0}$ and $k$ are positive constants.

A marine biologist models the oxygen concentration, $C$ (in $\mathrm{mg} / \mathrm{L}$ ), in a coral reef as a function of time, $t$ (in hours), over a 24 -hour period. The data suggests a sinusoidal pattern:

The biologist proposes a model of the form $C=a \sin (b t)+d$, where $a, b$, and $d$ are constants.

An ecologist models the population density of a species, $D$ (in individuals per $\mathrm{km}^{2}$ ), as a function of distance, $x$ (in km), from a pollution source. The data suggests a Gaussian distribution:

The ecologist proposes a model of the form $D=a e^{-b(x-c)^{2}}$, where $a, b$, and $c$ are constants.

An engineer models the vibration amplitude, $A$ (in mm), of a machine as a function of its operating frequency, $f$ (in Hz). The data suggests a polynomial relationship:

The engineer proposes a cubic model of the form $A=a f^{3}+b f^{2}+c f+d$, where $a, b, c$, and $d$ are constants.

A physicist investigates the cooling of a metal rod after being removed from a furnace. The temperature, $T$ (in ${ }^{\circ} \mathrm{C}$ ), of the rod at time $t$ (in minutes) is recorded as:

The physicist proposes a modified exponential model of the form $T=a e^{-k t}+T_{0}$, where $a, k$, and $T_{0}$ are constants, and $T_{0}$ represents the ambient temperature (assumed to be $20^{\circ} \mathrm{C}$ ).

A researcher studies the intensity of light, $I$ (in lumens), emitted by a bioluminescent organism as a function of time, $t$ (in hours), after exposure to a stimulus. The data is:

The researcher proposes a model of the form $I=a e^{-k t}+b$, where $a, k$, and $b$ are constants, and $b$ represents the baseline intensity.

A researcher is studying the relationship between the number of hours spent studying and the scores obtained in a test. The data collected is as follows: (2, 50), (3, 55), (5, 65), (7, 70), (8, 75), (10, 80).

A researcher is studying the relationship between the amount of fertilizer used (in kg) and the yield of a particular crop (in tons). The data collected is as follows: (2.3, 4.5), (3.1, 5.2), (4.7, 6.8), (5.5, 7.1), (6.2, 8.0), (7.4, 9.3). The researcher believes that a non-linear model would best describe the relationship between these variables.

A research team is evaluating two models to estimate the growth of a bacterial culture $G$ over time $t$ in hours.