In an automobile factory, certain metal parts are covered with a thin layer of nickel that protects them against corrosion and wear. The process used is electroplating with nickel.
It is assumed that the random variable $X$, which associates to each treated part the thickness of nickel deposited, follows a normal distribution with mean $\mu _ { 1 } = 25$ micrometers ( $\mu \mathrm { m }$ ) and standard deviation $\sigma _ { 1 }$.
A part is compliant if the thickness of nickel deposited is between $22.8 \mu \mathrm {~m}$ and $27.2 \mu \mathrm {~m}$.
The probability density function of $X$ is represented below. It was determined that $P ( X > 27.2 ) = 0.023$.

1. a. Determine the probability that a part is compliant. b. Justify that 1.1 is an approximate value of $\sigma _ { 1 }$ to within $10 ^ { - 1 }$. c. Given that a part is compliant, calculate the probability that the thickness of nickel deposited on it is less than $24 \mu \mathrm {~m}$. Round to $10 ^ { - 3 }$.
2. A team of engineers proposes another nickel plating process, obtained by chemical reaction without any current source. The team claims that this new process theoretically allows obtaining $98 \%$ of compliant parts. The random variable $Y$ which, for each part treated with this new process, associates the thickness of nickel deposited follows a normal distribution with mean $\mu _ { 2 } = 25 \mu \mathrm {~m}$ and standard deviation $\sigma _ { 2 }$. a. Assuming the above claim, compare $\sigma _ { 1 }$ and $\sigma _ { 2 }$. b. A quality control evaluates the new process; it reveals that out of 500 parts tested, 15 are not compliant. At the $95 \%$ confidence level, can we reject the team's claim?

Exercise 3

All requested results will be rounded to the nearest thousandth.

1. 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 $1.2\mathrm{~g/L}$.
   
2. 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.
   
3. 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?

The mass in grams of melons from market gardener B is modelled by a random variable $M_\mathrm{B}$ that follows a normal distribution with mean 1050 and unknown standard deviation $\sigma$. Melons are described as ``compliant'' if their mass is between 900 g and 1200 g. The retailer observes that $85\%$ of melons supplied by market gardener B are compliant. Determine the standard deviation $\sigma$ of the random variable $M_\mathrm{B}$. Give the value rounded to the nearest integer.

The number of trees per hectare in this forest can be modelled by a random variable $X$ following a normal distribution with mean $\mu = 4000$ and standard deviation $\sigma = 300$.

1. Determine the probability that there are between 3400 and 4600 trees on a given hectare of this forest. The result will be given rounded to $10^{-3}$.
2. Calculate the probability that there are more than 4500 trees on a given hectare of this forest. The result will be given rounded to $10^{-3}$.

Part of the city's DVD stock consists of animated films intended for young audiences. An animated film is chosen at random and we denote by $X$ the random variable that gives the duration, in minutes, of this film. $X$ follows a normal distribution with mean $\mu = 80$ min and standard deviation $\sigma$. Furthermore, it is estimated that $P(X \geqslant 92) = 0{,}10$.

1. Determine the real number $\sigma$ and give an approximate value to 0.01.
2. A child watches an animated film whose duration he does not know. Knowing that he has already watched one hour and thirty minutes, what is the probability that the film ends within the next five minutes?

A factory manufactures tubes.
Part A
Questions 1. and 2. are independent. We are interested in two types of tubes, called type 1 tubes and type 2 tubes.

1. A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres. a. Let $X$ denote the random variable that, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07.
   A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection. b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X _ { 1 }$ denote the random variable that, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X _ { 1 }$ follows a normal distribution with mean 1.5 and standard deviation $\sigma _ { 1 }$.
   A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10 ^ { - 3 }$ of $\sigma _ { 1 }$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac { X _ { 1 } - 1.5 } { \sigma _ { 1 } }$ which follows the standard normal distribution.)
   
2. A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2 \%$ of tubes that are not ``compliant for length'' is acceptable.
   It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''. a. Give an asymptotic confidence interval at $95 \%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes. b. Should the machine be serviced? Justify your answer.

Part B
Adjustment errors in the production line can affect the thickness or length of type 2 tubes.
A study conducted on the production revealed that:

• $96 \%$ of type 2 tubes have compliant thickness;
• among type 2 tubes that have compliant thickness, $95 \%$ have compliant length;
• $3.6 \%$ of type 2 tubes have non-compliant thickness and compliant length.

A type 2 tube is randomly selected from the production and we consider the events: --- $E$: ``the thickness of the tube is compliant''; --- $L$: ``the length of the tube is compliant''. We model the random experiment with a probability tree.

1. Copy and complete this tree entirely.
2. Show that the probability of event $L$ equals 0.948.

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 $8.2 \mathrm{~mm}$.

1. Calculate the probability that a bearing is compliant.
2. 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.
3. 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.

In this exercise and unless otherwise stated, results should be rounded to $10^{-3}$.
A factory manufactures tubes.

Part A

Questions 1. and 2. are independent. We are interested in two types of tubes, called type 1 tubes and type 2 tubes.

1. A type 1 tube is accepted at inspection if its thickness is between 1.35 millimetres and 1.65 millimetres. a. Let $X$ denote the random variable which, for each type 1 tube randomly selected from the day's production, gives its thickness expressed in millimetres. We assume that the random variable $X$ follows a normal distribution with mean 1.5 and standard deviation 0.07.
   A type 1 tube is randomly selected from the day's production. Calculate the probability that the tube is accepted at inspection. b. The company wishes to improve the quality of type 1 tube production. To do this, the settings of the machines producing these tubes are modified. Let $X_1$ denote the random variable which, for each type 1 tube selected from the production of the modified machine, gives its thickness. We assume that the random variable $X_1$ follows a normal distribution with mean 1.5 and standard deviation $\sigma_1$.
   A type 1 tube is randomly selected from the production of the modified machine. Determine an approximate value to $10^{-3}$ of $\sigma_1$ so that the probability that this tube is accepted at inspection equals 0.98. (You may use the random variable $Z$ defined by $Z = \frac{X_1 - 1.5}{\sigma_1}$ which follows the standard normal distribution.)
   
2. A machine produces type 2 tubes. A type 2 tube is said to be ``compliant for length'' when its length, in millimetres, belongs to the interval [298; 302]. The specifications establish that, in the production of type 2 tubes, a proportion of $2\%$ of tubes that are not ``compliant for length'' is acceptable.
   It is desired to decide whether the production machine should be serviced. To do this, a random sample of 250 tubes is taken from the production of type 2 tubes, in which 10 tubes are found to be not ``compliant for length''. a. Give an asymptotic confidence interval at $95\%$ for the frequency of tubes not ``compliant for length'' in a sample of 250 tubes. b. Should the machine be serviced? Justify your answer.

Part B

Adjustment errors in the production line can affect the thickness or length of type 2 tubes.
A study conducted on the production revealed that: --- $96\%$ of type 2 tubes have compliant thickness; --- among type 2 tubes that have compliant thickness, $95\%$ have compliant length; --- $3.6\%$ of type 2 tubes have non-compliant thickness and compliant length.
A type 2 tube is randomly selected from the production and we consider the events: --- $E$: ``the tube's thickness is compliant''; --- $L$: ``the tube's length is compliant''.
We model the random experiment with a probability tree.

1. Copy and complete this tree entirely.
2. Show that the probability of event $L$ equals 0.948.

During a professional examination, each candidate must present a file of type A or a file of type B; $60\%$ of candidates present a file of type A, the others presenting a file of type B. The jury assigns to each file a mark between 0 and 20. A candidate passes if the mark assigned to their file is greater than or equal to 10. A file is chosen at random. It is admitted that the mark assigned to a file of type A can be modeled by a random variable $X$ following the normal distribution with mean 11.3 and standard deviation 3, and the mark assigned to a file of type B by a random variable $Y$ following the normal distribution with mean 12.4 and standard deviation 4.7. We may denote $A$ the event: ``the file is a file of type A'', $B$ the event: ``the file is a file of type B'', and $R$ the event: ``the file is that of a candidate who passed the examination''. Probabilities will be rounded to the nearest hundredth.

1. The chosen file is of type A. What is the probability that this file is that of a candidate who passed the examination? It is admitted that the probability that the chosen file, given that it is of type B, is that of a candidate who passed is equal to 0.70.
2. Show that the probability, rounded to the nearest hundredth, that the chosen file is that of a candidate who passed the examination is equal to 0.68.
3. The jury examines 500 files chosen randomly from files of type B. Among these files, 368 are those of candidates who passed the examination.
   A jury member claims that this sample is not representative. He justifies his claim by explaining that in this sample, the proportion of candidates who passed is too large. What argument can be put forward to confirm or contest his claims?
4. The jury awards a ``jury prize'' to files that obtained a mark greater than or equal to $N$, where $N$ is an integer. The probability that a file chosen at random obtains the ``jury prize'' is between 0.10 and 0.15. Determine the integer $N$.

The maximum thickness of an avalanche, expressed in centimetres, can be modelled by a random variable $X$ which follows a normal distribution with mean $\mu = 150 \mathrm{~cm}$ and unknown standard deviation. We know that $P ( X \geqslant 200 ) = 0.025$. What is the probability $P ( X \geqslant 100 )$ ? a. We cannot b. 0.025 c. 0.95 d. 0.975 answer because there are missing elements in the problem statement.

Part B

The travel time for Louise, in minutes, between her home and work, can be modeled by a random variable $X$ that follows a normal distribution with mean 28 and standard deviation 5.

1. Calculate $P ( X \leqslant 25 )$.
2. Calculate the probability that the travel time is between 18 and 38 minutes.
3. Determine the travel duration $d$, rounded to the nearest minute, such that $P ( X \geqslant d ) = 0.1$.
4. Louise has now found a faster route. From now on, the travel time, in minutes, can be modeled by a random variable $Y$ that follows a normal distribution with mean 26 and standard deviation $\sigma$. We know that $P ( Y \geqslant 30 ) = 0.1$. Determine $\sigma$ rounded 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 machine manufactures balls intended for a game of chance. The mass in grams of each of these balls can be modeled by a random variable $M$ following a normal distribution with mean 52 and standard deviation $\sigma$. Balls whose mass is between 51 and 53 grams are said to be compliant.

1. With the initial settings of the machine we have $\sigma = 0.437$. Under these conditions, calculate the probability that a ball manufactured by this machine is compliant. An approximate value to $10 ^ { - 1 }$ near the result will be given.
2. It is considered that the machine is correctly adjusted if at least $99 \%$ of the balls it manufactures are compliant. Determine an approximate value of the largest value of $\sigma$ that allows us to affirm that the machine is correctly adjusted.

Consider $T$ the random variable following the normal distribution with mean $\mu = 60$ and standard deviation $\sigma = 6$.
The probability $P _ { ( T > 60 ) } ( T > 72 )$ rounded to the nearest thousandth is: Answer A: 0.954 Answer B: 1 Answer C: 0.023 Answer D: 0.046

All probabilities, unless otherwise stated, will be rounded to $10 ^ { - 3 }$ in this exercise. ``Chikungunya virus, transmitted to humans by the bite of the tiger mosquito, causes patients acute joint pain that can be persistent. In 2005, a major chikungunya epidemic affected the islands of the Indian Ocean and notably the island of Réunion, with several hundred thousand reported cases. In 2007, the disease appeared in Europe, then at the end of 2013, in the Caribbean and reached the American continent in 2014''. (https ://www.pasteur.fr/fr/centre-medical/fiches-maladies/chikungunya) A test has been developed for the detection of this virus. The laboratory manufacturing this test provides the following characteristics:

• the probability that an individual affected by the virus has a positive test is 0.999;
• the probability that an individual not affected by the virus has a positive test is 0.005.

A systematic screening test is carried out in a target population. An individual is chosen at random from this population. We call:

• $M$ the event: ``the chosen individual is affected by chikungunya''.
• $T$ the event: ``the test of the chosen individual is positive''.

The test is considered reliable when the probability that an individual with a positive test is affected by the virus is greater than 0.95.

Part A: Study of an example

1. Give the probabilities $P _ { M } ( T )$ and $P _ { \bar { M } } ( T )$. ``In March 2005, the epidemic spread rapidly on the island of Réunion, with a major surge between late April and early June followed by persistence of viral transmission during the austral winter. In total, 270,000 people were infected out of a total population of 750,000 individuals''. (https ://www.pasteur.fr/fr/centre-medical/fiches-maladies/chikungunya) At the end of 2005, the laboratory conducted a mass screening test of the population of the island of Réunion. In this part, the target population is therefore the population of the island of Réunion.
2. Give the exact value of $P ( M )$.
3. Copy and complete the weighted tree given below. [Figure]
4. Calculate the probability that an individual is affected by the virus and has a positive test.
5. Calculate the probability that an individual has a positive test.
6. Calculate the probability that an individual with a positive test is affected by the virus.
7. Can we estimate that this test is reliable? Argue.

Part B: Screening on a target population

In this part, we denote by $p$ the proportion of people affected by chikungunya virus in a target population. We seek here to test the reliability of this laboratory's test as a function of $p$.

1. Copy, adapting it, the weighted tree from question A3 taking into account the new data.
2. Express the probability $P ( T )$ as a function of $p$.
3. Show that $P _ { T } ( M ) = \frac { 999 p } { 994 p + 5 }$.
4. For which values of $p$ can we consider that this test is reliable?

Part C: Study on a sample

During the epidemic, it is admitted that the probability of being affected by chikungunya on the island of Réunion is 0.36. We consider a sample of $n$ individuals chosen at random, assimilating this choice to a random draw with replacement. We denote by $X$ the random variable counting the number of infected individuals in this sample among the $n$ drawn. It is admitted that $X$ follows a binomial distribution with parameters $n$ and $p = 0.36$. Determine from how many individuals $n$ the probability of the event ``at least one of the $n$ inhabitants of this sample is affected by the virus'' is greater than 0.99. Explain the approach.

The Body Mass Index (BMI) can be considered a practical, easy and inexpensive alternative for direct measurement of body fat. Its value can be obtained by the formula $\text{BMI} = \frac{\text{Mass}}{(\text{Height})^{2}}$, in which mass is in kilograms and height is in meters. Children naturally begin life with a high body fat index, but become thinner as they age, so scientists created a BMI especially for children and young adults, from two to twenty years of age, called BMI by age.
The graph shows the BMI by age for boys.
A mother decided to calculate the BMI of her son, a ten-year-old boy, with 1.20 m height and $30.92 \mathrm{~kg}$.
To be in the range considered normal for BMI, the minimum and maximum values that this boy needs to lose weight, in kilograms, should be, respectively,
(A) 1.12 and 5.12.
(B) 2.68 and 12.28.
(C) 3.47 and 7.47.
(D) 5.00 and 10.76.
(E) 7.77 and 11.77.

Consider the quadratic equation $x ^ { 2 } + b x + c = 0$, where $b$ and $c$ are chosen randomly from the interval $[ 0,1 ]$ with the probability uniformly distributed over all pairs $( b , c )$. Let $p ( b ) =$ the probability that the given equation has a real solution for given (fixed) value of $b$. Answer the following questions by filling in the blanks. a) The equation $x ^ { 2 } + b x + c = 0$ has a real solution if and only if $b ^ { 2 } - 4 c$ is
Answer: $\_\_\_\_$ b) The value of $p \left( \frac { 1 } { 2 } \right)$, i.e., the probability that $x ^ { 2 } + \frac { x } { 2 } + c = 0$ has a real solution is
Answer: $\_\_\_\_$ c) As a function of $b$, is $p ( b )$ increasing, decreasing or constant?
Answer: $\_\_\_\_$ d) As $b$ and $c$ both vary, what is the probability that $x ^ { 2 } + b x + c = 0$ has a real solution?
Answer: $\_\_\_\_$

The following table shows the results of a survey on customer preferences for hiking boots by manufacturer at a certain department store.

Manufacturer	A	B	C	D	Total
Preference (\%)	20	28	25	27	100

When 192 customers each purchase one pair of hiking boots, what is the probability that 42 or more customers will choose product C, using the standard normal distribution table on the right? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.6915
(2) 0.7745
(3) 0.8256
(4) 0.8332
(5) 0.8413

A music club is preparing for its regular concert this year. Based on past experience, the attendance rate among invited guests is 0.5. When 100 people are randomly selected from the invited guests,

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
1.0	0.3413
1.2	0.3849
1.4	0.4192
1.6	0.4452

what is the probability that the attendance rate is at least 0.43 and at most 0.56, using the standard normal distribution table on the right? [4 points]
(1) 0.8041
(2) 0.7698
(3) 0.7605
(4) 0.7262
(5) 0.6826

Suppose the weight of products produced at a certain factory follows a normal distribution $\mathrm { N } \left( 11,2 ^ { 2 } \right)$. Two people, $A$ and $B$, each independently extracted a sample of size 4. Using the standard normal distribution table on the right, what is the probability that the sample means extracted by both $A$ and $B$ are between 10 and 14 inclusive? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
1	0.3413
2	0.4772
3	0.4987

(1) 0.8123
(2) 0.7056
(3) 0.6587
(4) 0.5228
(5) 0.2944

The weight of products manufactured at a certain factory follows a normal distribution $\mathrm { N } \left( 11,2 ^ { 2 } \right)$. Two people A and B each independently randomly extracted a sample of size 4. Using the standard normal distribution table on the right, what is the probability that the sample means of both A and B are between 10 and 14 inclusive? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
1	0.3413
2	0.4772
3	0.4987

(1) 0.8123
(2) 0.7056
(3) 0.6587
(4) 0.5228
(5) 0.2944

At a certain car wash, the time required to wash one car follows a normal distribution with mean 30 minutes and standard deviation 2 minutes. When washing one car at this car wash, what is the probability that the washing time is 33 minutes or more, using the standard normal distribution table below? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.0228
(2) 0.0668
(3) 0.1587
(4) 0.2708
(5) 0.3085

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

A physical examination was conducted on 1000 new employees of a company, and the results show that height follows a normal distribution with mean $m$ and standard deviation 10. Among all new employees, 242 had a height of 177 cm or more. Using the standard normal distribution table on the right, find the probability that a randomly selected new employee from all new employees has a height of 180 cm or more. (Here, height is measured in 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

Let $\bar { X }$ be the sample mean of a sample of size 25 randomly extracted from a population that follows a normal distribution with population mean 75 and population standard deviation 5. For a random variable $Z$ following the standard normal distribution, a positive constant $c$ satisfies
$$\mathrm { P } ( | Z | > c ) = 0.06$$
Which of the following in are correct? [4 points]
ㄱ. For a constant $a$ such that $\mathrm { P } ( Z > a ) = 0.05$, we have $c > a$. ㄴ. $\mathrm { P } ( \bar { X } \leqq c + 75 ) = 0.97$ ㄷ. For a constant $b$ such that $\mathrm { P } ( \bar { X } > b ) = 0.01$, we have $c < b - 75$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ