\section*{Exercise 1 (5 points)}

A large cosmetics brand launches a new moisturizing cream.

\section*{Part A: Packaging of jars}
This brand wishes to sell the new cream in a 50 mL package and has jars with a maximum capacity of 55 mL for this purpose.\\
A jar of cream is said to be non-compliant if it contains less than 49 mL of cream.

\begin{enumerate}
  \item Several series of tests lead to modeling the quantity of cream, expressed in mL, contained in each jar by a random variable $X$ which follows the normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 1.2$.\\
Calculate the probability that a jar of cream is non-compliant.
  \item The proportion of non-compliant jars of cream is judged to be too large. By modifying the viscosity of the cream, we can change the value of the standard deviation of the random variable $X$, without modifying its mean $\mu = 50$. We want to reduce to 0.06 the probability that a randomly chosen jar is non-compliant.\\
We denote $\sigma ^ { \prime }$ the new standard deviation, and $Z$ the random variable equal to $\frac { X - 50 } { \sigma ^ { \prime } }$\\
a. Specify the distribution followed by the random variable $Z$.\\
b. Determine an approximate value of the real number $u$ such that $p ( Z \leqslant u ) = 0.06$.\\
c. Deduce the expected value of $\sigma ^ { \prime }$.
  \item A shop orders 50 jars of this new cream from its supplier.
\end{enumerate}

We consider that the work on the viscosity of the cream has made it possible to achieve the set objective and therefore that the proportion of non-compliant jars in the sample is 0.06.\\
Let $Y$ be the random variable equal to the number of non-compliant jars among the 50 jars received.\\
a. We admit that $Y$ follows a binomial distribution. Give its parameters.\\
b. Calculate the probability that the shop receives two non-compliant jars or fewer than two non-compliant jars.

\section*{Part B: Advertising campaign}
A consumer association decides to estimate the proportion of people satisfied by the use of this cream.\\
It conducts a survey among people using this product. Out of 140 people interviewed, 99 declare themselves satisfied.\\
Estimate, by confidence interval at the 95\% threshold, the proportion of satisfied people among the users of the cream.