All requested results will be rounded to the nearest thousandth.
A study conducted on a population of men aged 35 to 40 years showed that the total cholesterol level in the blood, expressed in grams per liter, can be modeled by a random variable $T$ that follows a normal distribution with mean $\mu = 1.84$ and standard deviation $\sigma = 0.4$. a. Determine according to this model the probability that a subject randomly selected from this population has a cholesterol level between $1.04\mathrm{~g/L}$ and $2.64\mathrm{~g/L}$. b. Determine according to this model the probability that a subject randomly selected from this population has a cholesterol level greater than $1.2\mathrm{~g/L}$.
In order to test the effectiveness of a cholesterol-lowering drug, patients needing treatment agreed to participate in a clinical trial organized by a laboratory. In this trial, $60\%$ of patients took the drug for one month, the others taking a placebo (neutral tablet). We study the decrease in cholesterol level after the experiment. A decrease in this level is observed in $80\%$ of patients who took the drug. No decrease is observed in $90\%$ of people who took the placebo. A patient who participated in the experiment is randomly selected and we denote:
$M$ the event ``the patient took the drug'';
$B$ the event ``the patient's cholesterol level decreased''.
a. Translate the data from the statement using a probability tree. b. Calculate the probability of event $B$. c. Calculate the probability that a patient took the drug given that their cholesterol level decreased.
The laboratory that produces this drug announces that $30\%$ of patients who use it experience side effects. To test this hypothesis, a cardiologist randomly selects 100 patients treated with this drug. a. Determine the asymptotic confidence interval at the $95\%$ threshold for the proportion of patients undergoing this treatment and experiencing side effects. b. The study conducted on 100 patients counted 37 people experiencing side effects. What can we conclude? c. To estimate the proportion of users of this drug experiencing side effects, an independent organization conducts a study based on a confidence interval at the $95\%$ confidence level. This study results in an observed frequency of $37\%$ of patients experiencing side effects, and a confidence interval that does not contain the frequency $30\%$. What is the minimum sample size for this study?
\section*{Exercise 3}
All requested results will be rounded to the nearest thousandth.
\begin{enumerate}
\item A study conducted on a population of men aged 35 to 40 years showed that the total cholesterol level in the blood, expressed in grams per liter, can be modeled by a random variable $T$ that follows a normal distribution with mean $\mu = 1.84$ and standard deviation $\sigma = 0.4$.\\
a. Determine according to this model the probability that a subject randomly selected from this population has a cholesterol level between $1.04\mathrm{~g/L}$ and $2.64\mathrm{~g/L}$.\\
b. Determine according to this model the probability that a subject randomly selected from this population has a cholesterol level greater than $1.2\mathrm{~g/L}$.
\item In order to test the effectiveness of a cholesterol-lowering drug, patients needing treatment agreed to participate in a clinical trial organized by a laboratory.\\
In this trial, $60\%$ of patients took the drug for one month, the others taking a placebo (neutral tablet).\\
We study the decrease in cholesterol level after the experiment.\\
A decrease in this level is observed in $80\%$ of patients who took the drug.\\
No decrease is observed in $90\%$ of people who took the placebo. A patient who participated in the experiment is randomly selected and we denote:
\begin{itemize}
\item $M$ the event ``the patient took the drug'';
\item $B$ the event ``the patient's cholesterol level decreased''.
\end{itemize}
a. Translate the data from the statement using a probability tree.\\
b. Calculate the probability of event $B$.\\
c. Calculate the probability that a patient took the drug given that their cholesterol level decreased.
\item The laboratory that produces this drug announces that $30\%$ of patients who use it experience side effects.\\
To test this hypothesis, a cardiologist randomly selects 100 patients treated with this drug.\\
a. Determine the asymptotic confidence interval at the $95\%$ threshold for the proportion of patients undergoing this treatment and experiencing side effects.\\
b. The study conducted on 100 patients counted 37 people experiencing side effects.\\
What can we conclude?\\
c. To estimate the proportion of users of this drug experiencing side effects, an independent organization conducts a study based on a confidence interval at the $95\%$ confidence level.\\
This study results in an observed frequency of $37\%$ of patients experiencing side effects, and a confidence interval that does not contain the frequency $30\%$.\\
What is the minimum sample size for this study?
\end{enumerate}