\textbf{Exercise 3} (3 points)

\section*{Part A:}
A health control agency is interested in the number of bacteria of a certain type contained in fresh cream. It performs analyses on 10000 samples of 1 ml of fresh cream from the entire French production. The results are given in the table below:

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
\begin{tabular}{ l }
Number of bacteria \\
(in thousands) \\
\end{tabular} & $[100;120[$ & $[120;130[$ & $[130;140[$ & $[140;150[$ & $[150;160[$ & $[160;180[$ \\
\hline
Number of samples & 1597 & 1284 & 2255 & 1808 & 1345 & 1711 \\
\hline
\end{tabular}
\end{center}

Using a calculator, give an estimate of the mean and standard deviation of the number of bacteria per sample.

\section*{Part B:}
The agency then decides to model the number of bacteria studied (in thousands per ml) present in fresh cream by a random variable $X$ following the normal distribution with parameters $\mu = 140$ and $\sigma = 19$.

\begin{enumerate}
  \item a. Is this choice of modelling relevant? Argue.\\
b. We denote $p = P(X \geqslant 160)$. Determine the value of $p$ rounded to $10^{-3}$.

  \item During the inspection of a dairy, the health control agency analyzes a sample of 50 samples of 1 ml of fresh cream from the production of this dairy; 13 samples contain more than 160 thousand bacteria.\\
a. The agency declares that there is an anomaly in the production and that it can affirm it with a probability of 0.05 of being wrong. Justify its declaration.\\
b. Could it have affirmed it with a probability of 0.01 of being wrong?
\end{enumerate}