The supplier claims that, among the high-end padlocks, there are no more than $3\%$ of defective padlocks in his production. The manager of the hardware store wishes to verify the validity of this claim in his stock; for this purpose, he takes a random sample of 500 high-end padlocks, and finds 19 that are defective.
Does this check call into question the fact that the stock contains no more than $3\%$ of defective padlocks?
For this, you may use an asymptotic fluctuation interval at the $95\%$ threshold.

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.

Market gardener C claims that $80\%$ of the melons in his production are compliant (mass between 900 g and 1200 g). The retailer doubts this claim. He observes that out of 400 melons delivered by this market gardener during one week, only 294 are compliant. Is the retailer right to doubt the claim of market gardener C?

The operator claims that the density of firs in this communal forest is 1 fir for every 2 trees. On a plot, 106 firs were counted in a sample of 200 trees. Does this result call into question the operator's claim?

The municipality of a large city has a stock of DVDs. Among the $6\%$ of defective DVDs in the entire stock, $98\%$ are removed. It is also admitted that among the non-defective DVDs, $92\%$ are kept in stock; the others are removed.
One of the city's media libraries wonders whether the number of defective DVDs it possesses is not abnormally high. To do this, it performs tests on a sample of 150 DVDs from its own stock which is large enough for this sample to be treated as successive sampling with replacement. On this sample, 14 defective DVDs are detected.
The asymptotic fluctuation interval at the $95\%$ threshold is given by the formula $$\left[ p - 1{,}96 \frac{\sqrt{p(1-p)}}{\sqrt{n}} ; p + 1{,}96 \frac{\sqrt{p(1+p)}}{\sqrt{n}} \right]$$ where $n$ denotes the sample size and $p$ the proportion of individuals possessing the characteristic studied in this population. The validity conditions are: $n \geqslant 30$, $np \geqslant 5$, $n(1-p) \geqslant 5$.
Can we reject the hypothesis that in this media library, $6\%$ of DVDs are defective?

A company packages white sugar from two farms $U$ and $V$ in 1 kg packets. Throughout the exercise, results should be rounded, if necessary, to the nearest thousandth.

1. The company announces that $30\%$ of the packets of sugar bearing the label ``extra fine'' that it packages contain sugar from farm U. Before validating an order, a buyer wants to verify this announced proportion. He randomly selects 150 packets from the company's production of packets labeled ``extra fine''. Among these packets, 30 contain sugar from farm U. Does he have reason to question the company's announcement?
2. The following year, the company declares that it has modified its production. The buyer wishes to estimate the new proportion of packets of sugar from farm U among the packets bearing the label ``extra fine''. He randomly selects 150 packets from the company's production of packets labeled ``extra fine''. Among these packets $42\%$ contain sugar from farm U. Give a confidence interval, at the $95\%$ confidence level, for the new proportion of packets labeled ``extra fine'' containing sugar from farm U.

In a supermarket, a department manager wishes to develop the supply of organic products. To justify his approach, he claims to his supervisor that $75\%$ of customers buy organic products at least once a month.
The supervisor wishes to verify his claims. To do this, he organizes a survey at the store exit. Of 2000 people interviewed, 1421 respond that they consume organic products at least once a month.
At the $95\%$ confidence level, what can we think of the department manager's claim?

Part C

Louise's company states on its website that $35\%$ of its employees practice carpooling. A survey conducted within the company shows that out of 254 employees randomly selected, 82 practice carpooling. Does this survey call into question the information published by the company on its website?

A customer orders a batch of 400 flutes of $12.5 \mathrm { cL }$ and finds that 13 of them do not conform to the characteristics announced by the manufacturer. The sales manager had nevertheless assured him that $98 \%$ of the flutes sold by the company were compliant. Does the customer's batch allow, at a risk of $5 \%$, to question the sales manager's claim?

Give an event in the context of the problem whose probability can be calculated using the term $\sum _ { k = 0 } ^ { 25 } \binom { 200 } { k } \cdot 0,1 ^ { k } \cdot ( 1 - 0,1 ) ^ { 200 - k }$.

In this part of the task, Machine $B$ is examined more closely. In the manufacturer's specifications for Machine $B$, it states that 30\% underfills can be expected. The responsible machine operator has the suspicion that Machine $B$ works better than specified. With a sample of 200 bottles, he tests his suspicion. If at most 45 underfills occur in this sample, he assumes that the machine works better than specified.
Determine the probability with which the machine operator assumes that the machine works better, even though the machine actually produces underfills with a probability of $p = 0.3$.

The machine operator responsible for the filling machine suspects that the machine actually works better than stated. By choosing $H _ { 0 } : p \geq 0.3$ as the null hypothesis, he wants to test his suspicion with a sample of 100 bottles. The number of underfilled bottles in the sample is again assumed to be binomially distributed.
(1) Determine a decision rule appropriate to the null hypothesis at a significance level of $\alpha = 0.05$.
(2) Describe the Type II error in the given context.