4.13 Non-linear regressionIB Maths: Applications and Interpretation HL: Subtopic test

1. 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.

   (a)

   Using exponential regression on your GDC, which model fits the data?

   [1 mark]

   • A$N=3.50+1.38t$
   • B$N=1.38\times3.50^{t}$
   • C$N=3.50e^{1.38t}$
   • D$N=3.50\times1.38^{t}$

   (b)

   Use the exponential model $N=3.50\times1.38^{t}$ to predict the number of bacteria, in thousands, at $t=8$ hours.

   [1 mark]

   • A46 thousand
   • B62 thousand
   • C33 thousand
   • D31 thousand

   (c)

   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]

   Total for question 1: 4 marks

2. 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)

   What is the coefficient of determination $R^2$ for the linear regression model?

   [1 mark]

   • A$0.982$
   • B$0.964$
   • C$0.991$
   • D$0.018$

   (b)

   Which statement about $R^2=0.998$ for the quadratic model is correct?

   [1 mark]

   • AThe model predicts $99.8\%$ of the braking distances exactly
   • BThere is a $99.8\%$ chance that the model is correct
   • C$99.8\%$ of the variation in speed is accounted for by braking distance
   • D$99.8\%$ of the variation in braking distance is accounted for by the model

   (c)

   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]

   Total for question 2: 4 marks

3. 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.

   (a)

   Use your GDC to find the power regression model $T=a\times d^{\,b}$, giving $a$ and $b$ to 3 significant figures.

   [3 marks]

   (b)

   Uranus has $d=19.19$ AU. Use your model to predict its orbital period to the nearest year. State whether this is interpolation or extrapolation and comment on the reliability of your prediction, given $R^2=1.00$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The depth of water $h$ metres in a harbour is measured every two hours from midnight, where $t$ is the number of hours after midnight. At $t=0, 2, 4, 6, 8, 10, 12, 14$ the depths are $1.53, 2.56, 3.72, 3.81, 2.70, 1.55, 1.41, 2.33$. The depth is modelled by a sine function $h=a\sin(bt+c)+d$ with $t$ in hours and the angle in radians. Use your GDC.

   (a)

   Use sine regression on your GDC to find the values of $a$, $b$, $c$ and $d$ to 3 significant figures. Hence write down the amplitude and the period of the tide, and interpret the value of $d$.

   [6 marks]

   (b)

   (i) Use the model to find the depth at $t=17$.
   (ii) Ships can enter the harbour when $h\ge3.2$. Find, for $0\le t\le12$, the length of time for which $h\ge3.2$.
   (iii) The GDC gives $R^2=0.999$ for the sine model. Interpret this value.

   [6 marks]

   Total for question 4: 12 marks

End of questions