An institute conducts a survey to determine, in a given population, the proportion of people who are in favour of a territorial development project. A random sample of people from this population is interviewed, and one question is asked to each person.
Part A: Number of people who agree to answer the survey
We admit in this part that the probability that a person interviewed agrees to answer the question is equal to 0.6.

1. The survey institute interviews 700 people. We denote by $X$ the random variable corresponding to the number of people interviewed who agree to answer the question asked. a. What is the distribution of the random variable $X$? Justify the answer. b. What is the best approximation of $P(X \geqslant 400)$ among the following numbers? 0.92 0.93 0.94 0.95.
   
2. How many people must the institute interview at minimum to guarantee, with a probability greater than 0.9, that the number of people answering the survey is greater than or equal to 400?

Part B: Proportion of people in favour of the project in the population
In this part, we assume that $n$ people have answered the question, and we admit that these people constitute a random sample of size $n$ (where $n$ is a natural number greater than 50). Among these people, 29\% are in favour of the development project.

1. Give a confidence interval, at the 95\% confidence level, for the proportion of people who are in favour of the project in the total population.
   
2. Determine the minimum value of the integer $n$ so that the confidence interval, at the 95\% confidence level, has an amplitude less than or equal to 0.04.

Part C: Correction due to insincere responses
In this part, we assume that, among the surveyed people who agreed to answer the question asked, 29\% claim that they are in favour of the project. The survey institute also knows that some people are not sincere and answer the opposite of their true opinion. Based on experience, the institute estimates at 15\% the rate of insincere responses among the people who responded, and admits that this rate is the same regardless of the opinion of the person interviewed.
A file of a person who responded is randomly selected, and we define:

• $F$ the event ``the person is actually in favour of the project'';
• $\bar{F}$ the event ``the person is actually opposed to the project'';
• $A$ the event ``the person claims that they are in favour of the project'';
• $\bar{A}$ the event ``the person claims that they are opposed to the project''.

Thus, according to the data, we have $p(A) = 0.29$.

1. By interpreting the data in the statement, indicate the values of $P_F(A)$ and $P_{\bar{F}}(A)$.
   
2. We set $x = P(F)$. a. Reproduce on your paper and complete the probability tree. b. Deduce an equality satisfied by $x$.
   
3. Determine, among the people who responded to the survey, the proportion of those who are actually in favour of the project.

Improvements made to line A had the effect of increasing the proportion $p$ of defect-free components. To estimate this proportion, a random sample of 400 components manufactured by line A is taken. In this sample, the observed frequency of defect-free components is 0.92.

1. Determine a confidence interval for the proportion $p$ at the $95\%$ confidence level.
2. What should be the minimum sample size so that such a confidence interval has a maximum amplitude of 0.02?

The company wishes to estimate the proportion of people over 20 years old among its customers, at a confidence level of $95\%$, with an interval amplitude less than 0.05. It questions for this purpose a random sample of customers.
What is the minimum number of customers to question?
Answer a: 40 Answer b: 400 Answer c: 1600 Answer d: 20

The tourist office wishes to conduct a survey to estimate the proportion of clients satisfied with the services offered at the ski resort. For this, it uses a confidence interval of length 0.04 with a confidence level of 0.95. The number of clients to interview is: a. 50 b. 2500 c. 25 d. 625

[Probability and Statistics] A survey of 100 randomly selected people from city A regarding the safest mode of transportation found that 20 people chose express buses. Using this result, a 95\% confidence interval for the proportion of people who chose express buses was found to be $[ a , b ]$. For city B, a 95\% confidence interval is to be constructed for the proportion of people who think express buses are the safest mode of transportation based on a random sample of $n$ people. Find the minimum value of $n$ such that the maximum allowable sampling error of this confidence interval is at most $\frac { b - a } { 2 }$. [4 points]

We want to investigate the proportion of antibody possession for a specific disease among Korean adults. Let $p$ be the proportion of antibody possession in the population, and let $\hat { p }$ be the proportion of antibody possession in a sample of $n$ people randomly selected from the population. Find the minimum value of $n$ such that the probability that $| \hat { p } - p | \leq 0.16 \sqrt { \hat { p } ( 1 - \hat { p } ) }$ is at least 0.9544. (where $Z$ is a random variable following the standard normal distribution and $\mathrm { P } ( 0 \leq Z \leq 2 ) = 0.4772$.) [4 points]

To determine the proportion of residents in a certain city who have experience using the central park, $n$ residents of the city were randomly sampled and surveyed. The result showed that 80\% had experience using the central park. Using this result, the 95\% confidence interval for the proportion of residents in the entire city who have experience using the central park is $[ a , b ]$. When $b - a = 0.098$, find the value of $n$. (Here, when $Z$ is a random variable following the standard normal distribution, calculate using $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95$.) [4 points]

A company produces shampoo where the volume of 1 bottle follows a normal distribution $\mathrm { N } \left( m , \sigma ^ { 2 } \right)$. A sample of 16 bottles was randomly selected from the company's shampoo, and the 95\% confidence interval for $m$ using the sample mean is $746.1 \leq m \leq 755.9$. When a sample of $n$ bottles is randomly selected and a 99\% confidence interval for $m$ is constructed as $a \leq m \leq b$, what is the minimum natural number $n$ such that $b - a \leq 6$? (Here, the unit of volume is mL, and when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95$ and $\mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [3 points]
(1) 70
(2) 74
(3) 78
(4) 82
(5) 86