The company ``Bonne Mamie'' uses a machine to fill jam jars on a production line. We denote by $X$ the random variable that associates to each jar of jam produced the mass of jam it contains, expressed in grams.\\
In the case where the machine is correctly adjusted, we admit that $X$ follows a normal distribution with mean $\mu = 125$ and standard deviation $\sigma$.

\begin{enumerate}
  \item a. For any positive real number $t$, determine a relationship between
$$P ( X \leqslant 125 - t ) \text { and } P ( X \geqslant 125 + t ) .$$
b. We know that $2.3\%$ of the jam jars contain less than 121 grams of jam. Using the previous relationship, determine
$$P ( 121 \leqslant X \leqslant 129 ) .$$
  \item Determine a value rounded to the nearest unit of $\sigma$ such that
$$P ( 123 \leqslant X \leqslant 127 ) = 0.68 .$$
\end{enumerate}

In the rest of the exercise, we assume that $\boldsymbol { \sigma } = \mathbf { 2 }$.

\begin{enumerate}
  \setcounter{enumi}{2}
  \item We estimate that a jar of jam is compliant when the mass of jam it contains is between 120 and 130 grams.\\
a. We randomly choose a jar of jam from the production. Determine the probability that this jar is compliant. The result will be given rounded to $10 ^ { - 4 }$.\\
b. We randomly choose a jar from those with a jam mass less than 130 grams. What is the probability that this jar is not compliant? The result will be given rounded to $10 ^ { - 4 }$.
  \item We admit that the probability, rounded to $10 ^ { - 3 }$, that a jar of jam is compliant is 0.988.\\
We randomly choose 900 jars from the production. We observe that 871 of these jars are compliant. At the 95\% threshold, can we reject the following hypothesis: ``The machine is correctly adjusted''?
\end{enumerate}