4.13 Non-linear regressionIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 4.13 Non-linear regression

1. The number of bacteria $N$ (in thousands) in a culture is counted every hour for six hours. At times $t=0, 1, 2, 3, 4, 5, 6$ hours the counts are $3.5, 4.8, 6.7, 9.2, 12.7, 17.5, 24.2$. Use your GDC.
   Interpret the value $1.38$ in the model $N=3.50\times1.38^{t}$, and state the link between this model and a geometric sequence.2 marks
2. The braking distance $d$ metres of a car was measured at speeds $v$ km/h. The results for $(v, d)$ were $(20, 6.4)$, $(30, 9.5)$, $(40, 16.9)$, $(50, 23.0)$, $(60, 34.6)$, $(70, 43.8)$, $(80, 58.7)$. Using your GDC, linear regression gives $r=0.982$, and quadratic regression gives $d=0.00940v^2-0.0719v+3.88$ with sum of squared residuals $SS_{res}=4.67$ and $R^2=0.998$.
   A student says: "The quadratic model has a larger $R^2$ than the linear model ($0.998>0.964$), so it will give a reliable braking distance at $150$ km/h." Give two reasons why this conclusion is not justified.2 marks
3. Astronomers record the mean distance $d$ of each planet from the Sun, in astronomical units (AU), and its orbital period $T$ in years: Mercury $(0.387, 0.241)$, Venus $(0.723, 0.615)$, Earth $(1.000, 1.000)$, Mars $(1.524, 1.881)$, Jupiter $(5.203, 11.86)$, Saturn $(9.537, 29.46)$. A power model $T=a\times d^{\,b}$ is to be fitted. Use your GDC.
   Use your GDC to find the power regression model $T=a\times d^{\,b}$, giving $a$ and $b$ to 3 significant figures.3 marks