A statistical study was conducted in a large city in France between January $1^{\text{st}}$ 2000 and January $1^{\text{st}}$ 2010 to assess the proportion of households with a fixed internet connection.\\
On January $1^{\text{st}}$ 2000, one household in eight had a fixed internet connection and, on January $1^{\text{st}}$ 2010, $64\%$ of households did.\\
Following this study, this proportion was modelled by the function $g$ defined on the interval $[0; +\infty[$ by:
$$g(t) = \frac{1}{1 + k\mathrm{e}^{-at}},$$
where $k$ and $a$ are two positive real constants and the variable $t$ denotes the time, measured in years, elapsed since January $1^{\text{st}}$ 2000.

\begin{enumerate}
  \item Determine the exact values of $k$ and $a$ so that $g(0) = \frac{1}{8}$ and $g(10) = \frac{64}{100}$.
  \item In the following, we take $k = 7$ and $a = 0.25$. The function $g$ is therefore defined by:
$$g(t) = \frac{1}{1 + 7\mathrm{e}^{-\left(\frac{t}{4}\right)}}$$
a. Show that the function $g$ is increasing on the interval $[0; +\infty[$.\\
b. According to this model, can we assert that one day, at least $99\%$ of households in this city will have a fixed internet connection? Justify your answer.\\
  \item a. Give, to the nearest hundredth, the proportion of households, predicted by the model, equipped with a fixed internet connection on January $1^{\text{st}}$ 2018.\\
b. Given the development of mobile telephony, some statisticians believe that the modelling by the function $g$ of the evolution of the proportion of households with a fixed internet connection should be reconsidered.\\
At the beginning of 2018 a survey was conducted. Out of 1000 households, 880 had a fixed connection.\\
Does this survey support these sceptical statisticians?\\
(You may use an asymptotic confidence interval at the $95\%$ level.)
\end{enumerate}