In this exercise and unless otherwise stated, results should be rounded to $10^{-3}$.

A factory manufactures tubes.

\section*{Part A}
Questions 1. and 2. are independent.\\
We are interested in two types of tubes, called type 1 tubes and type 2 tubes.

\begin{enumerate}
  \item A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres.\\
a. Let $X$ denote the random variable which, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07.

A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection.\\
b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X_1$ denote the random variable which, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X_1$ follows a normal distribution with mean 1.5 and standard deviation $\sigma_1$.

A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10^{-3}$ of $\sigma_1$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac{X_1 - 1.5}{\sigma_1}$ which follows the standard normal distribution.)

  \item A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2\%$ of tubes that are not ``compliant for length'' is acceptable.

It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''.\\
a. Give an asymptotic confidence interval at $95\%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes.\\
b. Should the machine be serviced? Justify your answer.
\end{enumerate}

\section*{Part B}
Adjustment errors in the production line can affect the thickness or length of type 2 tubes.\\
A study conducted on the production revealed that:\\
--- $96\%$ of type 2 tubes have compliant thickness;\\
--- among type 2 tubes that have compliant thickness, $95\%$ have compliant length;\\
--- $3.6\%$ of type 2 tubes have non-compliant thickness and compliant length.

A type 2 tube is randomly selected from the production and we consider the events:\\
--- $E$: ``the tube's thickness is compliant'';\\
--- $L$: ``the tube's length is compliant''.

We model the random experiment with a probability tree.

\begin{enumerate}
  \item Copy and complete this tree entirely.
  \item Show that the probability of event $L$ equals 0.948.
\end{enumerate}