An industrial bakery uses a machine to manufacture loaves of country bread weighing on average 400 grams. To be sold to customers, these loaves must weigh at least 385 grams. A loaf whose mass is strictly less than 385 grams is non-marketable, a loaf whose mass is greater than or equal to 385 grams is marketable.\\
The mass of a loaf manufactured by the machine can be modeled by a random variable $X$ following the normal distribution with mean $\mu = 400$ and standard deviation $\sigma = 11$.

Probabilities will be rounded to the nearest thousandth.

\section*{Part A}
You may use the following table in which values are rounded to the nearest thousandth.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
$x$ & 380 & 385 & 390 & 395 & 400 & 405 & 410 & 415 & 420 \\
\hline
$P ( X \leqslant x )$ & 0,035 & 0,086 & 0,182 & 0,325 & 0,5 & 0,675 & 0,818 & 0,914 & 0,965 \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \item Calculate $P ( 390 \leqslant X \leqslant 410 )$.
  \item Calculate the probability $p$ that a loaf chosen at random from production is marketable.
  \item The manufacturer finds this probability $p$ too low. He decides to modify his production methods in order to vary the value of $\sigma$ without changing that of $\mu$.\\
For what value of $\sigma$ is the probability that a loaf is marketable equal to $96\%$ ? Round the result to the nearest tenth.\\
You may use the following result: when $Z$ is a random variable that follows the normal distribution with mean 0 and standard deviation 1, we have $P ( Z \leqslant - 1,751 ) \approx 0,040$.
\end{enumerate}

\section*{Part B}
The production methods have been modified with the aim of obtaining $96\%$ marketable loaves.\\
To evaluate the effectiveness of these modifications, a quality control is performed on a sample of 300 loaves manufactured.

\begin{enumerate}
  \item Determine the asymptotic confidence interval at the $95\%$ confidence level for the proportion of marketable loaves in a sample of size 300.
  \item Among the 300 loaves in the sample, 283 are marketable.
\end{enumerate}

In light of the confidence interval obtained in question 1, can we decide that the objective has been achieved?

\section*{Part C}
The baker uses an electronic scale. The operating time without malfunction, in days, of this electronic scale is a random variable $T$ that follows an exponential distribution with parameter $\lambda$.

\begin{enumerate}
  \item We know that the probability that the electronic scale does not malfunction before 30 days is 0,913. Deduce the value of $\lambda$ rounded to the nearest thousandth.
\end{enumerate}

Throughout the rest, we will take $\lambda = 0,003$.\\
2. What is the probability that the electronic scale continues to function without malfunction after 90 days, given that it has functioned without malfunction for 60 days?\\
3. The seller of this electronic scale assured the baker that there was a one in two chance that the scale would not malfunction before a year. Is he right? If not, for how many days is this true?