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.

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 red-billed tropicbird is a bird of intertropical regions.
1. When the red-billed tropicbird lives in a polluted environment, its lifespan, in years, is modelled by a random variable $X$ following a normal distribution with unknown mean $\mu$ and standard deviation $\sigma = 0.95$.
a. Consider the random variable $Y$ defined by $Y = \frac { X - \mu } { 0.95 }$.
Give without justification the distribution followed by the variable $Y$.
b. It is known that $P ( X \geqslant 4 ) = 0.146$.
Prove that the value of $\mu$ rounded to the nearest integer is 3.
2. When the red-billed tropicbird lives in a healthy environment, its lifespan, in years, is modelled by a random variable $Z$.
The curves of the density functions associated with the distributions of $X$ and $Z$ are represented in the APPENDIX to be returned with the answer sheet.
a. Which is the curve of the density function associated with $X$? Justify.
b. On the APPENDIX to be returned with the answer sheet, shade the region of the plane corresponding to $P ( Z \geqslant 4 )$.
It will be admitted henceforth that $P ( Z \geqslant 4 ) = 0.677$.
3. A statistical study of a given region established that $30\%$ of red-billed tropicbirds live in a polluted environment; the others live in a healthy environment.
A red-billed tropicbird living in the given region is chosen at random.
Consider the following events:

• $S$ : ``the red-billed tropicbird chosen lives in a healthy environment'';
• $V$ : ``the red-billed tropicbird chosen has a lifespan of at least 4 years''.

a. Complete the weighted tree illustrating the situation on the APPENDIX to be returned with the answer sheet.
b. Determine $P ( V )$. Round the result to the nearest thousandth.
c. Given that the red-billed tropicbird has a lifespan of at least 4 years, what is the probability that it lives in a healthy environment? Round the result to the nearest thousandth.

A physical examination was conducted on 1000 new employees of a company, and it was found that height follows a normal distribution with mean $m$ and standard deviation 10. Among all new employees, 242 had a height of 177 or more. Using the standard normal distribution table on the right, what is the probability that a randomly selected new employee from all new employees has a height of 180 or more? (Here, the unit of height is cm.) [4 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.7	0.2580
0.8	0.2881
0.9	0.3159
1.0	0.3413

(1) 0.1587
(2) 0.1841
(3) 0.2119
(4) 0.2267
(5) 0.2420

The random variable $X$ follows a normal distribution with mean $m$ and standard deviation 5, and the probability density function $f ( x )$ of the random variable $X$ satisfies the following conditions. (가) $f ( 10 ) > f ( 20 )$ (나) $f ( 4 ) < f ( 22 )$ When $m$ is a natural number, $\mathrm { P } ( 17 \leq X \leq 18 ) = a$. Find the value of $1000a$ using the standard normal distribution table below. [4 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.6	0.226
0.8	0.288
1.0	0.341
1.2	0.385
1.4	0.419

The random variable $X$ follows a normal distribution with mean $m$ and standard deviation 5, and the probability density function $f ( x )$ of the random variable $X$ satisfies the following conditions.
(a) $f ( 10 ) > f ( 20 )$
(b) $f ( 4 ) < f ( 22 )$ When $m$ is a natural number, what is the value of $\mathrm { P } ( 17 \leq X \leq 18 )$ obtained using the standard normal distribution table on the right? [4 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.6	0.226
0.8	0.288
1.0	0.341
1.2	0.385
1.4	0.419

(1) 0.044
(2) 0.053
(3) 0.062
(4) 0.078
(5) 0.097