In a factory, two machines A and B are used to manufacture parts.

Machine A ensures $40\%$ of production and machine B ensures $60\%$. It is estimated that $10\%$ of parts from machine A have a defect and that $9\%$ of parts from machine B have a defect.

A part is chosen at random and we consider the following events:
\begin{itemize}
  \item $A$: ``The part is produced by machine A''
  \item $B$: ``The part is produced by machine B''
  \item $D$: ``The part has a defect''
  \item $\bar{D}$: the opposite event of event $D$.
\end{itemize}

\begin{enumerate}
  \item a. Translate the situation using a probability tree.\\
b. Calculate the probability that the chosen part has a defect and was manufactured by machine A.\\
c. Prove that the probability $P(D)$ of event $D$ is equal to 0.094.\\
d. It is observed that the chosen part has a defect. What is the probability that this part comes from machine A?
  \item It is estimated that machine A is properly adjusted if $90\%$ of the parts it manufactures are conforming. It is decided to check this machine by examining $n$ parts chosen at random ($n$ natural integer) from the production of machine A. These $n$ draws are treated as successive independent draws with replacement. We denote by $X_n$ the number of parts that are conforming in the sample of $n$ parts, and $F_n = \dfrac{X_n}{n}$ the corresponding proportion.\\
a. Justify that the random variable $X_n$ follows a binomial distribution and specify its parameters.\\
b. In this question, we take $n = 150$. Determine the asymptotic fluctuation interval $I$ at the $95\%$ threshold of the random variable $F_{150}$.\\
c. A quality test counts 21 non-conforming parts in a sample of 150 parts produced. Does this call into question the adjustment of the machine? Justify the answer.
\end{enumerate}