A study conducted in 2000 showed that the percentage of French people regularly consuming ice cream was 84\%. In 2010, out of 900 people surveyed, 795 of them declared consuming ice cream.
Can we affirm, at the 95\% confidence level and based on the study of this sample, that the percentage of French people regularly consuming ice cream remained stable between 2000 and 2010?

The store manager wishes to estimate the proportion of defective padlocks in his stock of budget padlocks. For this, he takes a random sample of 500 budget padlocks, among which 39 prove to be defective.
Give a confidence interval for this proportion at the $95\%$ confidence level.

A fair coin is flipped 100 times in succession. Which of the intervals below is an asymptotic fluctuation interval at the 95\% confidence level for the frequency of appearance of heads on this coin? a. $[ 0.371 ; 0.637 ]$ b. $[ 0.480 ; 0.523 ]$ c. [0.402; 0.598] d. $[ 0.412 ; 0.695 ]$

A company wishes to obtain an estimate of the proportion of people over 60 years old among its customers, at the 95\% confidence level, with an interval amplitude less than 0.05.
What is the minimum number of customers to survey? a. 400 b. 800 c. 1600 d. 3200

The company sought to improve the quality of its production and claims that there are no more than $6\%$ defective light bulbs in its production. A consumer association conducts a test on a sample and obtains 71 defective light bulbs out of 1000.

1. In the case where there would be exactly $6\%$ defective light bulbs, determine an asymptotic confidence interval at the $95\%$ level for the frequency of defective light bulbs in a random sample of size 1000.
2. Do we have reasons to question the company's claim?

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?

Part C - Satisfaction survey
The company announces a satisfaction rate of $85\%$ for its customers who called and reached an operator.
A consumer association wishes to verify this rate and surveys 1303 people. Among these, 1150 say they are satisfied. What do you think of the satisfaction rate announced by the company?

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

An electoral research institute receives an order in which the margin of error should be at most 2 percentage points (0.02).
The institute has 5 recent surveys, P1 to P5, on the subject of the order and will use the one with an error smaller than requested.
The data on the surveys are as follows:

Survey	$\boldsymbol{\sigma}$	$\boldsymbol{N}$	$\sqrt{\boldsymbol{N}}$
P1	0.5	1764	42
P2	0.4	784	28
P3	0.3	576	24
P4	0.2	441	21
P5	0.1	64	8

The error $e$ can be expressed by
$$|e| < 1.96 \frac{\sigma}{\sqrt{N}}$$
where $\sigma$ is a parameter and $N$ is the number of people interviewed by the survey. Which survey should be used?
(A) P1
(B) P2
(C) P3
(D) P4
(E) P5

At a certain factory, table tennis balls are dropped onto a steel floor from a fixed height, and the height to which the table tennis ball bounces follows a normal distribution. When 100 table tennis balls produced by this factory were randomly sampled and the bounce height was measured, the mean was 245 and the standard deviation was 20. What is the number of integers in the 95\% confidence interval for the mean bounce height of all table tennis balls produced by this factory? (Note: The unit of height is mm, and when $Z$ follows a standard normal distribution, $\mathrm { P } ( 0 \leqq Z \leqq 1.96 ) = 0.4750$.) [3 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

A factory produces table tennis balls. When dropped from a certain height onto a steel floor, the height to which the table tennis ball bounces follows a normal distribution. From the table tennis balls produced by this factory, 100 balls were randomly sampled and the bounce height was measured, resulting in a mean of 245 and a standard deviation of 20.
What is the number of integers in the 95\% confidence interval for the mean bounce height of all table tennis balls produced by this factory? (Here, the unit of height is mm, and when $Z$ follows the standard normal distribution, $\mathrm { P } ( 0 \leqq Z \leqq 1.96 ) = 0.4750$.) [3 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

A population follows a normal distribution $\mathrm { N } \left( m , 2 ^ { 2 } \right)$. Let $\overline { X _ { A } }$ and $\overline { X _ { B } }$ be the sample means of samples of size 7 and size 10, respectively, drawn from this population. Using the distributions of $\overline { X _ { A } }$ and $\overline { X _ { B } }$, the 95\% confidence intervals for the population mean $m$ are $[a, b]$ and $[c, d]$, respectively. Choose all correct statements from the given options. [4 points]
Options ㄱ. The variance of $\overline { X _ { A } }$ is greater than the variance of $\overline { X _ { B } }$. ㄴ. $\mathrm { P } \left( \overline { X _ { A } } \leqq m + 2 \right) < \mathrm { P } \left( \overline { X _ { B } } \leqq m + 2 \right)$ ㄷ. $d - c < b - a$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

(Probability and Statistics) To determine the proportion $p$ of students arriving before 8 AM at a certain high school, 300 students were randomly sampled from the school on a certain day, and the sample proportion $\hat{p}$ of students arriving before 8 AM was obtained. Using the sample proportion $\hat{p}$, the 95\% confidence interval for the proportion $p$ is $[ 0.701, 0.799 ]$. Find the number of students among the 300 randomly sampled students who arrived before 8 AM. (Note: When $Z$ follows the standard normal distribution, $\mathrm { P } ( | Z | \leqq 1.96 ) = 0.95$.) [4 points]

The calcium content in one bottle of beverage produced by a certain company follows a normal distribution with population mean $m$ and population standard deviation $\sigma$. When 16 bottles of beverage produced by this company were randomly sampled and the calcium content was measured, the sample mean was 12.34. When the 95\% confidence interval for the population mean $m$ of the calcium content in one bottle of beverage produced by this company is $11.36 \leq m \leq a$, what is the value of $a + \sigma$? (Note: When $Z$ follows the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 1.96 ) = 0.4750$, and the unit of calcium content is mg.) [3 points]
(1) 14.32
(2) 14.82
(3) 15.32
(4) 15.82
(5) 16.32

The lifespan of monitors produced by a certain company follows a normal distribution. From a random sample of 100 monitors produced by this company, the sample mean is $\bar{x}$ and the sample standard deviation is 500. Using this result, the confidence interval for the mean lifespan of monitors produced by this company at a confidence level of $95\%$ is $[\bar{x} - c, \bar{x} + c]$. Find the value of $c$. (Here, $Z$ is a random variable following the standard normal distribution, and $\mathrm{P}(0 \leq Z \leq 1.96) = 0.4750$.) [3 points]

For a normal distribution with known standard deviation $\sigma$, a sample of size $n$ is randomly extracted from the population. The 95\% confidence interval for the population mean obtained from this sample is [100.4, 139.6]. Using the same sample, find the number of natural numbers contained in the 99\% confidence interval for the population mean. (Given that when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 1.96 ) = 0.475$ and $\mathrm { P } ( 0 \leq Z \leq 2.58 ) = 0.495$.) [3 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]

The weight of pomegranates produced at a certain farm follows a normal distribution with mean $m$ and standard deviation 40. A sample of size 64 was taken from the pomegranates produced at this farm, and the sample mean of the pomegranate weights was $\bar { x }$. Using this result, the 99\% confidence interval for the mean $m$ of the pomegranate weights produced at this farm is $\bar { x } - c \leq m \leq \bar { x } + c$. What is the value of $c$? (Here, the unit of weight is g, and when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 2.58 ) = 0.495$.) [4 points]
(1) 25.8
(2) 21.5
(3) 17.2
(4) 12.9
(5) 8.6

The weight of watermelons harvested in a certain village follows a normal distribution with mean $m$ kg and standard deviation 1.4 kg. When 49 watermelons are randomly sampled from this village and a 95\% confidence interval for the mean weight $m$ is constructed using the sample mean, the interval is $a \leq m \leq 7.992$. What is the value of $a$? (Here, when $Z$ is a random variable following the standard normal distribution, calculate using $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95$.) [3 points]
(1) 7.198
(2) 7.208
(3) 7.218
(4) 7.228
(5) 7.238

The daily leisure activity time of residents in a certain region follows a normal distribution with mean $m$ minutes and standard deviation $\sigma$ minutes. When 16 residents are randomly sampled and the sample mean of daily leisure activity time is 75 minutes, the 95\% confidence interval for the population mean $m$ is $a \leq m \leq b$. When 16 residents are randomly sampled again and the sample mean of daily leisure activity time is 77 minutes, the 99\% confidence interval for the population mean $m$ is $c \leq m \leq d$. Find the value of $\sigma$ that satisfies $d - b = 3.86$. (Here, when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95 , \mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [4 points]

A certain automobile company produces electric vehicles whose driving range on a single charge follows a normal distribution with mean $m$ and standard deviation $\sigma$.
When a sample of 100 electric vehicles produced by this company is randomly selected and the sample mean of the driving range is $\overline { x _ { 1 } }$, the 95\% confidence interval for the population mean $m$ is $a \leq m \leq b$.
When a sample of 400 electric vehicles produced by this company is randomly selected and the sample mean of the driving range is $\overline { x _ { 2 } }$, the 99\% confidence interval for the population mean $m$ is $c \leq m \leq d$.
If $\overline { x _ { 1 } } - \overline { x _ { 2 } } = 1.34$ and $a = c$, what is the value of $b - a$? (Here, the unit of driving range is km, and when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95 , \mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [3 points]
(1) 5.88
(2) 7.84
(3) 9.80
(4) 11.76
(5) 13.72

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

From a population following a normal distribution $\mathrm{N}(m, 5^2)$, a sample of size 49 is randomly extracted, and the sample mean is $\bar{x}$. The 95\% confidence interval for the population mean $m$ is $a \leq m \leq \frac{6}{5}a$. Find the value of $\bar{x}$. (Here, if $Z$ is a random variable following the standard normal distribution, $\mathrm{P}(|Z| \leq 1.96) = 0.95$.) [3 points]
(1) 15.2
(2) 15.4
(3) 15.6
(4) 15.8
(5) 16.0

A sample of size 256 is randomly extracted from a population following a normal distribution $\mathrm{N}\left(m, 2^{2}\right)$. The 95\% confidence interval for $m$ obtained using the sample mean is $a \leq m \leq b$. What is the value of $b - a$? (Given: When $Z$ is a random variable following the standard normal distribution, $\mathrm{P}(|Z| \leq 1.96) = 0.95$.) [3 points]
(1) 0.49
(2) 0.52
(3) 0.55
(4) 0.58
(5) 0.61