A pharmaceutical laboratory offers screening tests for various diseases. Its communications department highlights the following characteristics:
the probability that a sick person tests positive is 0.99;
the probability that a healthy person tests positive is 0.001.
For a disease that has just appeared, the laboratory develops a new test. A statistical study makes it possible to estimate that the percentage of sick people among the population of a metropolis is equal to $0.1 \%$. A person is chosen at random from this population and undergoes the test. We denote by $M$ the event ``the chosen person is sick'' and $T$ the event ``the test is positive''. a. Translate the statement in the form of a weighted tree. b. Prove that the probability $p ( T )$ of event $T$ is equal to $$1.989 \times 10 ^ { - 3 } .$$ c. Is the following statement true or false? Justify your answer. Statement: ``If the test is positive, there is less than one chance in two that the person is sick''.
The laboratory decides to market a test as soon as the probability that a person who tests positive is sick is greater than or equal to 0.95. We denote by $x$ the proportion of people affected by a certain disease in the population. From what value of $x$ does the laboratory market the corresponding test?
Part B
The laboratory's production line manufactures, in very large quantities, tablets of a medicine.
A tablet is compliant if its mass is between 890 and 920 mg. We assume that the mass in milligrams of a tablet taken at random from production can be modeled by a random variable $X$ that follows the normal distribution $\mathscr { N } \left( \mu , \sigma ^ { 2 } \right)$, with mean $\mu = 900$ and standard deviation $\sigma = 7$. a. Calculate the probability that a tablet drawn at random is compliant. Round to $10 ^ { - 2 }$. b. Determine the positive integer $h$ such that $P ( 900 - h \leqslant X \leqslant 900 + h ) \approx 0.99$ to within $10 ^ { - 3 }$.
The production line has been adjusted to obtain at least $97 \%$ compliant tablets. To evaluate the effectiveness of the adjustments, a check is performed by taking a sample of 1000 tablets from production. The size of the production is assumed to be large enough that this sample can be treated as 1000 successive draws with replacement. The check made it possible to count 53 non-compliant tablets in the sample taken. Does this check call into question the adjustments made by the laboratory? An asymptotic fluctuation interval at the $95 \%$ threshold can be used.
Parts A and B can be treated independently.
\section*{Part A}
A pharmaceutical laboratory offers screening tests for various diseases. Its communications department highlights the following characteristics:
\begin{itemize}
\item the probability that a sick person tests positive is 0.99;
\item the probability that a healthy person tests positive is 0.001.
\end{itemize}
\begin{enumerate}
\item For a disease that has just appeared, the laboratory develops a new test. A statistical study makes it possible to estimate that the percentage of sick people among the population of a metropolis is equal to $0.1 \%$. A person is chosen at random from this population and undergoes the test.\\
We denote by $M$ the event ``the chosen person is sick'' and $T$ the event ``the test is positive''.\\
a. Translate the statement in the form of a weighted tree.\\
b. Prove that the probability $p ( T )$ of event $T$ is equal to
$$1.989 \times 10 ^ { - 3 } .$$
c. Is the following statement true or false? Justify your answer.\\
Statement: ``If the test is positive, there is less than one chance in two that the person is sick''.\\
\item The laboratory decides to market a test as soon as the probability that a person who tests positive is sick is greater than or equal to 0.95. We denote by $x$ the proportion of people affected by a certain disease in the population. From what value of $x$ does the laboratory market the corresponding test?
\end{enumerate}
\section*{Part B}
The laboratory's production line manufactures, in very large quantities, tablets of a medicine.
\begin{enumerate}
\item A tablet is compliant if its mass is between 890 and 920 mg. We assume that the mass in milligrams of a tablet taken at random from production can be modeled by a random variable $X$ that follows the normal distribution $\mathscr { N } \left( \mu , \sigma ^ { 2 } \right)$, with mean $\mu = 900$ and standard deviation $\sigma = 7$.\\
a. Calculate the probability that a tablet drawn at random is compliant. Round to $10 ^ { - 2 }$.\\
b. Determine the positive integer $h$ such that $P ( 900 - h \leqslant X \leqslant 900 + h ) \approx 0.99$ to within $10 ^ { - 3 }$.
\item The production line has been adjusted to obtain at least $97 \%$ compliant tablets. To evaluate the effectiveness of the adjustments, a check is performed by taking a sample of 1000 tablets from production. The size of the production is assumed to be large enough that this sample can be treated as 1000 successive draws with replacement.\\
The check made it possible to count 53 non-compliant tablets in the sample taken.\\
Does this check call into question the adjustments made by the laboratory? An asymptotic fluctuation interval at the $95 \%$ threshold can be used.
\end{enumerate}