\section*{Exercise 1 (5 points)}
\section*{Part A}
An oyster farmer raises two species of oysters: ``the flat'' and ``the Japanese''. Each year, flat oysters represent $15 \%$ of his production.\\
Oysters are said to be of size $\mathrm { n } ^ { \circ } 3$ when their mass is between 66 g and 85 g. Only $10 \%$ of flat oysters are of size $\mathrm { n } ^ { \circ } 3$, whereas $80 \%$ of Japanese oysters are.

\begin{enumerate}
  \item The health service randomly selects an oyster from the oyster farmer's production. We assume that all oysters have an equal chance of being selected.\\
We consider the following events:
\begin{itemize}
  \item J: ``the selected oyster is a Japanese oyster'',
  \item C: ``the selected oyster is of size $\mathrm { n } ^ { \circ } 3$''.
\end{itemize}
a. Construct a complete weighted tree representing the situation.\\
b. Calculate the probability that the selected oyster is a flat oyster of size $\mathrm { n } ^ { \mathrm { o } } 3$.\\
c. Justify that the probability of obtaining an oyster of size $\mathrm { n } ^ { \circ } 3$ is 0.695.\\
d. The health service selected an oyster of size $\mathrm { n } ^ { \circ } 3$. What is the probability that it is a flat oyster?

  \item The mass of an oyster can be modeled by a random variable $X$ following the normal distribution with mean $\mu = 90$ and standard deviation $\sigma = 2$.\\
a. Give the probability that the oyster selected from the oyster farmer's production has a mass between 87 g and 89 g.\\
b. Give $\mathrm { P } ( \mathrm { X } \geqslant 91 )$.
\end{enumerate}

\section*{Part B}
This oyster farmer claims that $60 \%$ of his oysters have a mass greater than 91 g.\\
A restaurant owner would like to buy a large quantity of oysters but would first like to verify the oyster farmer's claim.

The restaurant owner purchases 10 dozen oysters from this oyster farmer, which we will consider as a sample of 120 oysters drawn at random. His production is large enough that we can treat it as sampling with replacement.\\
He observes that 65 of these oysters have a mass greater than 91 g.

\begin{enumerate}
  \item Let F be the random variable that associates to any sample of 120 oysters the frequency of those with a mass greater than 91 g.\\
After verifying the conditions of application, give an asymptotic confidence interval at the $95 \%$ level for the random variable F.
  \item What can the restaurant owner think of the oyster farmer's claim?
\end{enumerate}