The standard inner diameter of a bearing on a roller wheel is 8 mm.\\
Let $X$ denote the random variable giving in mm the diameter of a bearing and we assume that $X$ follows a normal distribution with mean 8 and standard deviation 0.1.\\
A bearing is said to be compliant if its diameter is between $7.8 \mathrm{~mm}$ and $8.2 \mathrm{~mm}$.

\begin{enumerate}
  \item Calculate the probability that a bearing is compliant.
  \item Supplier $B$ sells its bearings in batches of 16 and claims that only $5\%$ of its bearings are non-compliant.\\
The club president, who bought 30 batches from him, finds that 38 bearings are non-compliant. Does this check call into question supplier B's claim?\\
An asymptotic fluctuation interval at the $95\%$ threshold may be used.
  \item The bearing manufacturer of this supplier decides to improve the production of its bearings. The adjustment of the machine that manufactures them is modified so that $96\%$ of the bearings are compliant. We assume that after adjustment the random variable $X$ follows a normal distribution with mean 8 and standard deviation $\sigma$.\\
a. What is the distribution followed by $\frac{X - 8}{\sigma}$?\\
b. Determine $\sigma$ so that the manufactured bearing is compliant with a probability equal to 0.96.
\end{enumerate}