Non-linear regression and causation

The three scatter plots in the diagram correspond to three different studies. For each of them, indicate which type of model seems most suitable — linear ($y = a + bx$), quadratic ($y = a + bx + cx^2$) or exponential ($y = a \cdot b^x$) — and briefly justify your choice.

The number of users of an application grows approximately exponentially: $y = a \cdot b^x$, where $x$ is the number of months elapsed. Taking natural logarithms, $\ln y = \ln a + x \cdot \ln b$, so that the pairs $(x, \ln y)$ fit a straight line. The linearised data are:

1. a) Verify that the four linearised points lie exactly on a straight line and find its equation.

1. b) Deduce the values of $a$ and $b$ (express $a$ and $b$ as powers of $e$ and approximate them to two decimal places).

1. c) Estimate the number of users after $6$ months.

The area of a lake covered by an invasive plant has been measured over $5$ months:

1. a) Plot the pairs $(x, \ln y)$, verify that they align reasonably well, and fit a straight line to the linearised data (take $\ln y$ to two decimal places).

1. b) Write down the corresponding exponential model $y = a \cdot b^x$ and calculate the monthly growth rate as a percentage.

1. c) Predict the area covered in month $8$ and, given that the lake has $1.500\ \text{m}^2$, comment on the problem presented by extrapolating this model in the long term.

In a Physics laboratory, the period $T$ of a pendulum is measured as a function of its length $L$, and a power relationship $T = a \cdot L^b$ is suspected. Taking logarithms, $\ln T = \ln a + b \cdot \ln L$. The linearised data are:

1. a) Fit a straight line to the linearised data and write down its equation.

1. b) Deduce the values of $a$ and $b$.

1. c) Compare the value obtained for $b$ with that predicted by Physics, $b = \frac{1}{2}$, and comment on the result.

It is suspected that the variable $y$ has a quadratic dependence on $x$. Using the five data items in the table

the sums $\sum x_i = 10$, $\sum x_i^2 = 30$, $\sum x_i^3 = 100$, $\sum x_i^4 = 354$, $\sum y_i = 25$, $\sum x_i y_i = 60$ and $\sum x_i^2 y_i = 204$ have been calculated.

1. a) Set up the system of three equations that the coefficients of the regression parabola $y = a x^2 + bx + c$ must satisfy.

1. b) Solve it and write down the equation of the quadratic fit.

1. c) Estimate the value of $y$ for $x = 5$ and comment on the reliability of this prediction.

The consumption of an electric car, in $\text{kWh}$ per $100\ \text{km}$, has been measured at different speeds and the data are shown in the figure:

1. a) Mentally represent (or sketch) the scatter plot and explain why a quadratic model is more suitable than a linear one.

1. b) Calculate the necessary sums $\sum x_i$, $\sum x_i^2$, $\sum x_i^3$, $\sum x_i^4$, $\sum y_i$, $\sum x_i y_i$, $\sum x_i^2 y_i$ and set up the system for the quadratic fit.

1. c) Solve the system and write down the equation of the regression parabola (round the coefficients to four decimal places).

1. d) Find the speed for minimum consumption according to the model and predict the consumption at $140\ \text{km/h}$, assessing the reliability of this prediction.

You can do it on paper and send a photo with your answer.

State whether the following statement is true or false, giving reasons: 'The correlation coefficient between ice cream sales and the number of drownings in swimming pools is high and positive; therefore, selling more ice cream causes more drownings.' If it is false, explain the misinterpretation error and name a variable that could explain both at the same time.

In several districts of a European region, a correlation $r = 0,85$ has been observed between the number of storks recorded and the number of births registered.

1. a) Explain why it would be absurd to conclude that storks 'bring' babies, identifying the type of correlation shown in this example.

1. b) Propose a plausible confounding variable that explains the correlation, and explain how it acts on both variables.

1. c) Briefly outline what data you would collect and what calculation you would perform to unmask the spurious relationship.