In this part, the calculated probabilities will be rounded to the nearest thousandth. The industrialist markets his valves to many customers. Monthly demand is a random variable $D$ that follows the normal distribution with mean $\mu = 800$ and standard deviation $\sigma = 40$.

1. Determine $P(760 \leqslant D \leqslant 840)$.
2. Determine $P(D \leqslant 880)$.
3. The industrialist thinks that if he builds a monthly stock of 880 valves, he will have no more than a $1\%$ chance of running out of stock. Is he right?

2. Let $Y$ denote the random variable which, for a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$, associates its sugar content. We assume that $Y$ follows the normal distribution with mean $m _ { 2 } = 0.17$ and standard deviation $\sigma _ { 2 }$. We further assume that the probability that a small jar randomly selected from the production of line $\mathrm { F } _ { 2 }$ is compliant equals 0.99 . Let Z be the random variable defined by $Z = \frac { Y - m _ { 2 } } { \sigma _ { 2 } }$. a. What distribution does the random variable $Z$ follow? b. Determine, as a function of $\sigma _ { 2 }$, the interval to which $Z$ belongs when $Y$ belongs to the interval $[ 0.16 ; 0.18 ]$. c. Deduce an approximate value to $10 ^ { - 3 }$ near of $\sigma _ { 2 }$.
You may use the table given below, in which the random variable $Z$ follows the normal distribution with mean 0 and standard deviation 1 .

$\beta$	$P ( - \beta \leqslant Z \leqslant \beta )$
2.4324	0.985
2.4573	0.986
2.4838	0.987
2.5121	0.988
2.5427	0.989
2.5758	0.990
2.6121	0.991
2.6521	0.992
2.6968	0.993

Exercise 3

Common to all candidates

Given a real number $k$, we consider the function $f _ { k }$ defined on $\mathbb { R }$ by
$$f _ { k } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - k x } }$$
The plane is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ).

Part A

In this part we choose $k = 1$. We have therefore, for all real $x , f _ { 1 } ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - x } }$. The graph $\mathscr { C } _ { 1 }$ of the function $f _ { 1 }$ in the coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ) is given in APPENDIX, to be returned with your answer sheet.

1. Determine the limits of $f _ { 1 } ( x )$ as $x \to + \infty$ and as $x \to - \infty$ and give a graphical interpretation of the results obtained.
2. Prove that, for all real $x , f _ { 1 } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }$.
3. Let $f _ { 1 } ^ { \prime }$ denote the derivative function of $f _ { 1 }$ on $\mathbb { R }$. Calculate, for all real $x , f _ { 1 } ^ { \prime } ( x )$.

Deduce the variations of the function $f _ { 1 }$ on $\mathbb { R }$.

An industrial bakery uses a machine to manufacture loaves of country bread weighing on average 400 grams. To be sold to customers, these loaves must weigh at least 385 grams. A loaf whose mass is strictly less than 385 grams is non-marketable, a loaf whose mass is greater than or equal to 385 grams is marketable. The mass of a loaf manufactured by the machine can be modeled by a random variable $X$ following the normal distribution with mean $\mu = 400$ and standard deviation $\sigma = 11$.
Probabilities will be rounded to the nearest thousandth.

Part A

You may use the following table in which values are rounded to the nearest thousandth.

$x$	380	385	390	395	400	405	410	415	420
$P ( X \leqslant x )$	0,035	0,086	0,182	0,325	0,5	0,675	0,818	0,914	0,965

1. Calculate $P ( 390 \leqslant X \leqslant 410 )$.
2. Calculate the probability $p$ that a loaf chosen at random from production is marketable.
3. The manufacturer finds this probability $p$ too low. He decides to modify his production methods in order to vary the value of $\sigma$ without changing that of $\mu$. For what value of $\sigma$ is the probability that a loaf is marketable equal to $96\%$ ? Round the result to the nearest tenth. You may use the following result: when $Z$ is a random variable that follows the normal distribution with mean 0 and standard deviation 1, we have $P ( Z \leqslant - 1,751 ) \approx 0,040$.

Part B

The production methods have been modified with the aim of obtaining $96\%$ marketable loaves. To evaluate the effectiveness of these modifications, a quality control is performed on a sample of 300 loaves manufactured.

1. Determine the asymptotic confidence interval at the $95\%$ confidence level for the proportion of marketable loaves in a sample of size 300.
2. Among the 300 loaves in the sample, 283 are marketable.

In light of the confidence interval obtained in question 1, can we decide that the objective has been achieved?

Part C

The baker uses an electronic scale. The operating time without malfunction, in days, of this electronic scale is a random variable $T$ that follows an exponential distribution with parameter $\lambda$.

1. We know that the probability that the electronic scale does not malfunction before 30 days is 0,913. Deduce the value of $\lambda$ rounded to the nearest thousandth.

Throughout the rest, we will take $\lambda = 0,003$.
2. What is the probability that the electronic scale continues to function without malfunction after 90 days, given that it has functioned without malfunction for 60 days?
3. The seller of this electronic scale assured the baker that there was a one in two chance that the scale would not malfunction before a year. Is he right? If not, for how many days is this true?

A factory manufactures spherical balls whose diameter is expressed in millimetres. A ball is said to be out of specification when its diameter is less than 9 mm or greater than 11 mm.
Part A
1. Let $X$ be the random variable that associates to each ball chosen at random from production its diameter expressed in mm.
It is assumed that the random variable $X$ follows the normal distribution with mean 10 and standard deviation 0.4.
Show that an approximate value to 0.0001 of the probability that a ball is out of specification is 0.0124. You may use the table of values given in the appendix.
2. A production control is put in place such that 98\% of out-of-specification balls are rejected and 99\% of correct balls are kept.
A ball is chosen at random from production. Let $N$ denote the event: ``the chosen ball is within specification'', and $A$ the event: ``the chosen ball is accepted after the control''.
a. Construct a weighted tree diagram that incorporates the data from the problem statement.
b. Calculate the probability of event $A$.
c. What is the probability that an accepted ball is out of specification?
Part B
This production control proving too costly for the company, it is abandoned: henceforth, all balls produced are kept, and they are packaged in bags of 100 balls.
It is considered that the probability that a ball is out of specification is 0.0124.
It will be assumed that taking a bag of 100 balls at random is equivalent to performing a sampling with replacement of 100 balls from the set of manufactured balls.
Let $Y$ be the random variable that associates to every bag of 100 balls the number of out-of-specification balls in that bag.
1. What is the distribution followed by the random variable $Y$?
2. What are the mean and standard deviation of the random variable $Y$?
3. What is the probability that a bag of 100 balls contains exactly two out-of-specification balls?
4. What is the probability that a bag of 100 balls contains at most one out-of-specification ball?

Exercise 1 (5 points)

A large cosmetics brand launches a new moisturizing cream.

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.

1. 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.
2. 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 $\sigma ^ { \prime }$.
3. A shop orders 50 jars of this new cream from its supplier.

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.

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.

A football is compliant with regulations if it meets, depending on its size, two conditions simultaneously (on its mass and on its circumference). In particular, a standard-sized football is compliant with regulations when its mass, expressed in grams, belongs to the interval [410;450] and its circumference, expressed in centimetres, belongs to the interval [68;70].

1. Let $X$ denote the random variable which, for each standard-sized football chosen at random in the company, associates its mass in grams. It is admitted that $X$ follows a normal distribution with mean 430 and standard deviation 10. Determine an approximate value to $10 ^ { - 3 }$ of the probability $P ( 410 \leqslant X \leqslant 450 )$.
2. Let $Y$ denote the random variable which, for each standard-sized football chosen at random in the company, associates its circumference in centimetres. It is admitted that $Y$ follows a normal distribution with mean 69 and standard deviation $\sigma$. Determine the value of $\sigma$, to the nearest hundredth, knowing that $97 \%$ of standard-sized footballs have a circumference compliant with regulations. You may use the following result: when $Z$ is a random variable that follows the standard normal distribution, then $P ( - \beta \leqslant Z \leqslant \beta ) = 0,97$ for $\beta \approx 2,17$.

Each cone is filled with vanilla ice cream. We denote by $Y$ the random variable which, to each cone, associates the mass (expressed in grams) of ice cream it contains. It is assumed that $Y$ follows a normal distribution $\mathscr{N}\left(110 ; \sigma^{2}\right)$, with mean $\mu = 110$ and standard deviation $\sigma$.
An ice cream is considered marketable when the mass of ice cream it contains belongs to the interval $[104; 116]$.
Determine an approximate value to $10^{-1}$ of the parameter $\sigma$ such that the probability of the event ``the ice cream is marketable'' is equal to 0.98.

We assume in this part that the student uses the bicycle to go to his school. When he uses the bicycle, his travel time, expressed in minutes, between his home and his school is modeled by a random variable $T$ which follows a normal distribution with mean $\mu = 17$ and standard deviation $\sigma = 1.2$.

1. Determine the probability that the student takes between 15 and 20 minutes to get to his school.
2. He leaves his home by bicycle at 7:40 a.m. What is the probability that he is late for school?
3. The student leaves by bicycle. Before what time must he leave to arrive on time at school with a probability of 0.9? Round the result to the nearest minute.

When the student uses the bus, his travel time, expressed in minutes, between his home and his school is modeled by a random variable $T'$ which follows a normal distribution with mean $\mu' = 15$ and standard deviation $\sigma'$. We know that the probability that it takes him more than 20 minutes to get to his school by bus is 0.05. We denote by $Z'$ the random variable equal to $\frac{T' - 15}{\sigma'}$

1. What distribution does the random variable $Z'$ follow?
2. Determine an approximate value to 0.01 of the standard deviation $\sigma'$ of the random variable $T'$.

Question 4

This hypermarket sells baguettes of bread whose mass, expressed in grams, is a random variable that follows a normal distribution with mean 200 g. The probability that the mass of a baguette is between 184 g and 216 g is equal to 0.954. The probability that a baguette chosen at random has a mass less than 192 g has a value rounded to the nearest hundredth of: a. 0.16 b. 0.32 c. 0.84 d. 0.48

Part B

Between two phases of the competition, to improve, the competitor works on his lateral precision on another training target. He shoots arrows to try to hit a vertical band, with width 20 cm (shaded in the figure), as close as possible to the central vertical line. The plane containing the vertical band is equipped with a coordinate system: the central line aimed at is the $y$-axis. Let $X$ denote the random variable that, for any arrow shot reaching this plane, associates the abscissa of its point of impact.
It is assumed that the random variable $X$ follows a normal distribution with mean 0 and standard deviation 10.

1. When the arrow reaches the plane, determine the probability that its point of impact is located outside the shaded band.
2. How should the edges of the shaded band be modified so that, when the arrow reaches the plane, its point of impact is located inside the band with a probability equal to 0.6?

A factory produces mineral water in bottles. When the calcium level in a bottle is less than $6.5 \mathrm { mg }$ per litre, the water in that bottle is said to be very low in calcium.
Let $X$ be the random variable that, for each bottle randomly selected from the daily production of source A, associates the calcium level of the water it contains. We assume that $X$ follows a normal distribution with mean 8 and standard deviation 1.6. Let $Y$ be the random variable that, for each bottle randomly selected from the daily production of source B, associates the calcium level it contains. We assume that $Y$ follows a normal distribution with mean 9 and standard deviation $\sigma$.

1. Determine the probability that the calcium level measured in a bottle randomly taken from the daily production of source A is between $6.4 \mathrm { mg }$ and $9.6 \mathrm { mg }$.
2. Calculate the probability $p ( X \leqslant 6.5 )$.
3. Determine $\sigma$ knowing that the probability that a bottle randomly selected from the daily production of source B contains water that is very low in calcium is 0.1.

According to a statistical study conducted over several months, it is assumed that the number $X$ of budget padlocks sold per month in the hardware store can be modeled by a random variable that follows a normal distribution with mean $\mu = 750$ and standard deviation $\sigma = 25$.
Calculate $P(725 \leqslant X \leqslant 775)$.

According to a statistical study conducted over several months, it is assumed that the number $X$ of budget padlocks sold per month in the hardware store can be modeled by a random variable that follows a normal distribution with mean $\mu = 750$ and standard deviation $\sigma = 25$.
The store manager wants to know the number $n$ of budget padlocks he must have in stock at the beginning of the month, so that the probability of running out of stock during the month is less than 0.05. The stock is not replenished during the month.
Determine the smallest integer value of $n$ satisfying this condition.

The incubation time, expressed in hours, of the virus can be modeled by a random variable $T$ following a normal distribution with standard deviation $\sigma = 10$. We wish to determine its mean $\mu$.

1. a. Conjecture, using the graph of the probability density function, an approximate value of $\mu$. b. We are given $P(T < 110) = 0.18$. Shade on the graph a region whose area corresponds to the given probability.
2. We denote by $T'$ the random variable equal to $\frac{T - \mu}{10}$. a. What distribution does the random variable $T'$ follow? b. Determine an approximate value to the nearest unit of the mean $\mu$ of the random variable $T$ and verify the conjecture from question 1.

Let $X$ be a random variable that follows the normal distribution with mean $\mu = 110$ and standard deviation $\sigma = 25$. What is the value rounded to the nearest thousandth of the probability $P ( X \geqslant 135 )$ ? a. 0.159 b. 0.317 c. 0.683 d. 0.841

In a country, the height in centimetres of women aged 18 to 65 can be modelled by a random variable $X _ { 1 }$ following a normal distribution with mean $\mu _ { 1 } = 165 \mathrm {~cm}$ and standard deviation $\sigma _ { 1 } = 6 \mathrm {~cm}$, and that of men aged 18 to 65 by a random variable $X _ { 2 }$ following a normal distribution with mean $\mu _ { 2 } = 175 \mathrm {~cm}$ and standard deviation $\sigma _ { 2 } = 11 \mathrm {~cm}$. In this exercise all results should be rounded to $10 ^ { - 2 }$.

1. What is the probability that a woman chosen at random in this country measures between 1.53 metres and 1.77 metres?
2. a. Determine the probability that a man chosen at random in this country measures more than 1.70 metres. b. Furthermore, it is known that in this country women represent $52 \%$ of the population of people aged between 18 and 65. A person aged between 18 and 65 is chosen at random. They measure more than $1.70 \mathrm {~m}$. What is the probability that this person is a woman?

Exercise 3 -- Common to all candidates

Part A: Study of the lifespan of a household appliance

Statistical studies have made it possible to model the lifespan, in months, of a type of dishwasher by a random variable $X$ following a normal distribution $\mathscr{N}(\mu, \sigma^2)$ with mean $\mu = 84$ and standard deviation $\sigma$. Furthermore, we have $P(X \leqslant 64) = 0.16$.

1. a. By exploiting the graph, determine $P(64 \leqslant X \leqslant 104)$. b. What approximate integer value of $\sigma$ can we propose?
2. We denote by $Z$ the random variable defined by $Z = \dfrac{X - 84}{\sigma}$. a. What is the probability distribution followed by $Z$? b. Justify that $P(X \leqslant 64) = P\!\left(Z \leqslant \dfrac{-20}{\sigma}\right)$. c. Deduce the value of $\sigma$, rounded to $10^{-3}$.
3. In this question, we consider that $\sigma = 20.1$.
   The probabilities requested will be rounded to $10^{-3}$. a. Calculate the probability that the lifespan of the dishwasher is between 2 and 5 years. b. Calculate the probability that the dishwasher has a lifespan greater than 10 years.

Part B: Study of the warranty extension offered by El'Ectro

The dishwasher is guaranteed free of charge for the first two years. The company El'Ectro offers its customers a warranty extension of 3 additional years. Statistical studies conducted on customers who take the warranty extension show that $11.5\%$ of them use the warranty extension.

1. We randomly choose 12 customers among those who have taken the warranty extension (this choice can be treated as random sampling with replacement given the large number of customers). a. What is the probability that exactly 3 of these customers use this warranty extension? Detail the approach by specifying the probability distribution used. Round to $10^{-3}$. b. What is the probability that at least 6 of these customers use this warranty extension? Round to $10^{-3}$.
2. The warranty extension offer is as follows: for 65 euros additional, El'Ectro will reimburse the customer the initial value of the dishwasher, namely 399 euros, if an irreparable breakdown occurs between the beginning of the third year and the end of the fifth year. The customer cannot use this warranty extension if the breakdown is repairable.
   We randomly choose a customer among those who have subscribed to the warranty extension, and we denote by $Y$ the random variable representing the algebraic gain in euros realized on this customer by the company El'Ectro, thanks to the warranty extension. a. Justify that $Y$ takes the values 65 and $-334$ then give the probability distribution of $Y$. b. Is this warranty extension offer financially advantageous for the company? Justify.

In an industrial bakery, a baguette is randomly selected from production. It is admitted that the random variable expressing its mass, in grams, follows the normal distribution with mean 200 and standard deviation 10.
Statement 1: The probability that the mass of the baguette is greater than 187 g is greater than 0.9.
Indicate whether this statement is true or false, justifying your answer.

A company manufactures spherical wooden balls using two production machines A and B. The company considers that a ball can be sold only when its diameter is between $0.9 \mathrm{~cm}$ and $1.1 \mathrm{~cm}$.
Parts A, B and C are independent.
Part A
A study of the operation of the machines made it possible to establish the following results:

• $96\%$ of daily production is saleable.
• Machine A provides $60\%$ of daily production.
• The proportion of saleable balls among the production of machine A is $98\%$.

A ball is chosen at random from the production of a given day. The following events are defined: $A$: ``the ball was manufactured by machine A''; $B$: ``the ball was manufactured by machine B''; $V$: ``the ball is saleable''.

1. Determine the probability that the chosen ball is saleable and comes from machine A.
2. Justify that $P(B \cap V) = 0.372$ and deduce the probability that the chosen ball is saleable given that it comes from machine B.
3. A technician claims that $70\%$ of non-saleable balls come from machine B. Is he correct?

Part B

1. A statistical study leads to modelling the diameter of a ball randomly selected from the production of machine B by a random variable $X$ which follows a normal distribution with mean $\mu = 1$ and standard deviation $\sigma = 0.055$. Verify that the probability that a ball produced by machine B is saleable is indeed that found in Part A, to the nearest hundredth.
2. In the same way, the diameter of a ball randomly selected from the production of machine A is modelled using a random variable $Y$ which follows a normal distribution with mean $\mu = 1$ and standard deviation $\sigma'$, $\sigma'$ being a strictly positive real number. Given that $P(0.9 \leqslant Y \leqslant 1.1) = 0.98$, determine an approximate value to the nearest thousandth of $\sigma'$.

Part C
The saleable balls then pass through a machine that colours them randomly and with equal probability in white, black, blue, yellow or red. After being mixed, the balls are packaged in bags. The quantity produced is large enough that filling a bag can be treated as successive sampling with replacement of balls from daily production.

1. In this question only, the bags are all composed of 40 balls. a. A bag of balls is chosen at random. Determine the probability that the chosen bag contains exactly 10 black balls. Round the result to $10^{-3}$. b. In a bag of 40 balls, 12 black balls were counted. Does this observation allow us to question the adjustment of the machine that colours the balls?
2. If the company wishes the probability of obtaining at least one black ball in a bag to be greater than or equal to $99\%$, what is the minimum number of balls each bag must contain to achieve this objective?

A market gardener specializes in strawberry production.
Part A: strawberry production
The market gardener produces strawberries in two greenhouses denoted A and B; $55\%$ of strawberry flowers are in greenhouse A, and $45\%$ in greenhouse B. In greenhouse A, the probability that each flower produces fruit is equal to 0.88; in greenhouse B, it is equal to 0.84.
For each of the following propositions, indicate whether it is true or false by justifying the answer. An unjustified answer will not be taken into account.
Proposition 1: The probability that a strawberry flower, chosen at random from this farm, produces fruit is equal to 0.862.
Proposition 2: It is observed that a flower, chosen at random from this farm, produces fruit. The probability that it is located in greenhouse A, rounded to the nearest thousandth, is equal to 0.439.
Part B: strawberry packaging
Strawberries are packaged in trays. The mass (expressed in grams) of a tray can be modeled by a random variable $X$ which follows the normal distribution with mean $\mu = 250$ and standard deviation $\sigma$.

1. We are given $P ( X \leqslant 237 ) = 0.14$. Calculate the probability of the event ``the mass of the tray is between 237 and 263 grams''.
2. Let $Y$ be the random variable defined by: $Y = \frac { X - 250 } { \sigma }$. a. What is the distribution of the random variable $Y$? b. Prove that $P \left( Y \leqslant - \frac { 13 } { \sigma } \right) = 0.14$. c. Deduce the value of $\sigma$ rounded to the nearest integer.
3. In this question, we assume that $\sigma$ equals 12. We denote by $n$ and $m$ two integers. a. A tray is compliant if its mass, expressed in grams, lies in the interval $[ 250 - n ; 250 + n ]$. Determine the smallest value of $n$ for a tray to be compliant with a probability greater than or equal to $95\%$. b. In this question, we consider that a tray is compliant if its mass, expressed in grams, lies in the interval $[230; m ]$. Determine the smallest value of $m$ for a tray to be compliant with a probability greater than or equal to $95\%$.

Exercise 1 Common to all candidates

6 POINTS
The three parts are independent. Probability results should be rounded to $10^{-3}$ near.

Part 1

It is estimated that in 2013 the world population consists of 4.6 billion people aged 20 to 79 years and that $46.1\%$ of people aged 20 to 79 years live in rural areas and $53.9\%$ in urban areas. In 2013, according to the International Diabetes Federation, $9.9\%$ of the world population aged 20 to 79 years living in urban areas suffers from diabetes and $6.4\%$ of the world population aged 20 to 79 years living in rural areas suffers from diabetes.
A person aged 20 to 79 years is randomly selected. We denote: $R$ the event: ``the chosen person lives in a rural area'', $D$ the event: ``the chosen person suffers from diabetes''.

1. Translate this situation using a probability tree.
2. a. Calculate the probability that the interviewed person is diabetic. b. The chosen person is diabetic. What is the probability that they live in a rural area?

Part 2

A person is said to be hypoglycemic if their fasting blood glucose is less than $60 \mathrm{mg}.\mathrm{dL}^{-1}$ and they are hyperglycemic if their fasting blood glucose is greater than $110 \mathrm{mg}.\mathrm{dL}^{-1}$. Fasting blood glucose is considered ``normal'' if it is between $70 \mathrm{mg}.\mathrm{dL}^{-1}$ and $110 \mathrm{mg}.\mathrm{dL}^{-1}$. People with a blood glucose level between 60 and $70 \mathrm{mg}.\mathrm{rdL}^{-1}$ are not subject to special monitoring. An adult is randomly chosen from this population. A study established that the probability that they are hyperglycemic is 0.052 to $10^{-3}$ near. In the following, we will assume that this probability is equal to 0.052. We model the fasting blood glucose, expressed in $\mathrm{mg}.\mathrm{dL}^{-1}$, of an adult from a given population, by a random variable $X$ which follows a normal distribution with mean $\mu$ and standard deviation $\sigma$.

1. What is the probability that the chosen person has ``normal'' fasting blood glucose?
2. Determine the value of $\sigma$ rounded to the nearest tenth.
3. In this question, we take $\sigma = 12$. Calculate the probability that the chosen person is hypoglycemic.

Part 3

In order to estimate the proportion, for the year 2013, of people diagnosed with diabetes in the French population aged 20 to 79 years, 10000 people are randomly interviewed. In the sample studied, 716 people were diagnosed with diabetes.

1. Using a confidence interval at the $95\%$ confidence level, estimate the proportion of people diagnosed with diabetes in the French population aged 20 to 79 years.
2. What should be the minimum number of people to interview if we want to obtain a confidence interval with amplitude less than or equal to 0.01?

Statistical studies have made it possible to model the weekly time, in hours, of internet connection for young people in France aged 16 to 24 years by a random variable $T$ following a normal distribution with mean $\mu = 13.9$ and standard deviation $\sigma$.

1. We know that $p ( T \geqslant 22 ) = 0.023$.
   By exploiting this information: a. shade on the graph provided in the appendix, two distinct regions whose area is equal to 0.023; b. determine $P ( 5.8 \leqslant T \leqslant 22 )$. Justify the result. Show that an approximate value of $\sigma$ to one decimal place is 4.1.
2. A young person in France is chosen at random.
   Determine the probability that they are connected to the internet for more than 18 hours per week. Round to the nearest hundredth.

The company ``Bonne Mamie'' uses a machine to fill jam jars on a production line. We denote by $X$ the random variable that associates to each jar of jam produced the mass of jam it contains, expressed in grams. In the case where the machine is correctly adjusted, we admit that $X$ follows a normal distribution with mean $\mu = 125$ and standard deviation $\sigma$.

1. a. For any positive real number $t$, determine a relationship between $$P ( X \leqslant 125 - t ) \text { and } P ( X \geqslant 125 + t ) .$$ b. We know that $2.3\%$ of the jam jars contain less than 121 grams of jam. Using the previous relationship, determine $$P ( 121 \leqslant X \leqslant 129 ) .$$
2. Determine a value rounded to the nearest unit of $\sigma$ such that $$P ( 123 \leqslant X \leqslant 127 ) = 0.68 .$$

In the rest of the exercise, we assume that $\boldsymbol { \sigma } = \mathbf { 2 }$.

1. We estimate that a jar of jam is compliant when the mass of jam it contains is between 120 and 130 grams. a. We randomly choose a jar of jam from the production. Determine the probability that this jar is compliant. The result will be given rounded to $10 ^ { - 4 }$. b. We randomly choose a jar from those with a jam mass less than 130 grams. What is the probability that this jar is not compliant? The result will be given rounded to $10 ^ { - 4 }$.
2. We admit that the probability, rounded to $10 ^ { - 3 }$, that a jar of jam is compliant is 0.988. We randomly choose 900 jars from the production. We observe that 871 of these jars are compliant. At the 95\% threshold, can we reject the following hypothesis: ``The machine is correctly adjusted''?

Exercise 4 — Candidates who have NOT followed the specialization course
For each of the following statements, say whether it is true or false by justifying the answer. One point is awarded for each correct justified answer. An unjustified answer will not be taken into account and the absence of an answer is not penalized.

• In the diagram below, the density curve of a random variable $X$ following a normal distribution with mean $\mu = 20$ is represented. The probability that the random variable $X$ is between 20 and 21.6 is equal to 0.34.

Statement 1: The probability that the random variable $X$ belongs to the interval $[23.2; + \infty [$ is approximately 0.046.

• Let $z$ be a complex number different from 2. We set:

$$Z = \frac { \mathrm { i } z } { z - 2 }$$
Statement 2: The set of points in the complex plane with affixe $z$ such that $| Z | = 1$ is a line passing through point $\mathrm { A } ( 1 ; 0 )$. Statement 3: $Z$ is a pure imaginary number if and only if $z$ is real.

• Let $f$ be the function defined on $\mathbb { R }$ by:

$$f ( x ) = \frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 x } }$$
Statement 4: The equation $f ( x ) = 0.5$ has a unique solution on $\mathbb { R }$. Statement 5: The following algorithm displays as output the value 0.54.

\begin{tabular}{l} Variables: Initialization:

Processing:

Output:

&

$X$ and $Y$ are real numbers

$X$ takes the value 0

$Y$ takes the value $\frac { 3 } { 10 }$

While $Y < 0.5$

$X$ takes the value $X + 0.01$

$Y$ takes the value $\frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 X } }$

End While

Display $X$

\hline \end{tabular}