In July 2014, the health surveillance institute of an island published that $15\%$ of the population is affected by the virus. To verify whether the actual proportion is higher, a sample of 1000 people chosen at random from this island is studied. The population is large enough to consider that such a sample results from draws with replacement.

We denote by $X$ the random variable which, for any sample of 1000 people chosen at random, corresponds to the number of people affected by the virus and by $F$ the random variable giving the associated frequency.

\begin{enumerate}
  \item a. Under the hypothesis $p = 0.15$, determine the distribution of $X$.\\
  b. In a sample of 1000 people chosen at random from the island, 197 people affected by the virus are counted. What conclusion can be drawn from this observation about the figure of $15\%$ published by the health surveillance institute? Justify. (You may use the calculation of a fluctuation interval at the $95\%$ threshold.)
  \item We now consider that the value of $p$ is unknown. Using the sample from question 1.b., propose a confidence interval for the value of $p$, at the $95\%$ confidence level.
\end{enumerate}