Question 4

This hypermarket sells baguettes of bread whose mass, expressed in grams, is a random variable that follows a normal distribution with mean 200 g. The probability that the mass of a baguette is between 184 g and 216 g is equal to 0.954. The probability that a baguette chosen at random has a mass less than 192 g has a value rounded to the nearest hundredth of: a. 0.16 b. 0.32 c. 0.84 d. 0.48

Exercise 4 — Candidates who have NOT followed the specialization course
For each of the following statements, say whether it is true or false by justifying the answer. One point is awarded for each correct justified answer. An unjustified answer will not be taken into account and the absence of an answer is not penalized.

• In the diagram below, the density curve of a random variable $X$ following a normal distribution with mean $\mu = 20$ is represented. The probability that the random variable $X$ is between 20 and 21.6 is equal to 0.34.

Statement 1: The probability that the random variable $X$ belongs to the interval $[23.2; + \infty [$ is approximately 0.046.

• Let $z$ be a complex number different from 2. We set:

$$Z = \frac { \mathrm { i } z } { z - 2 }$$
Statement 2: The set of points in the complex plane with affixe $z$ such that $| Z | = 1$ is a line passing through point $\mathrm { A } ( 1 ; 0 )$. Statement 3: $Z$ is a pure imaginary number if and only if $z$ is real.

• Let $f$ be the function defined on $\mathbb { R }$ by:

$$f ( x ) = \frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 x } }$$
Statement 4: The equation $f ( x ) = 0.5$ has a unique solution on $\mathbb { R }$. Statement 5: The following algorithm displays as output the value 0.54.

\begin{tabular}{l} Variables: Initialization:

Processing:

Output:

&

$X$ and $Y$ are real numbers

$X$ takes the value 0

$Y$ takes the value $\frac { 3 } { 10 }$

While $Y < 0.5$

$X$ takes the value $X + 0.01$

$Y$ takes the value $\frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 X } }$

End While

Display $X$

\hline \end{tabular}

We study the production of a factory that manufactures sweets, packaged in bags. A bag is chosen at random from daily production. The mass of this bag, expressed in grams, is modelled by a random variable $X$ which follows a normal distribution with mean $\mu = 175$. Furthermore, statistical observation has shown that $2\%$ of bags have a mass less than or equal to 170 g, which is expressed in the model considered by: $P ( X \leqslant 170 ) = 0.02$.
What is the probability, rounded to the nearest hundredth, of the event ``the mass of the bag is between 170 and 180 grams''?
Answer a: 0.04 Answer b: 0.96 Answer c: 0.98 Answer d: We cannot answer because data is missing.

(Probability and Statistics) Random variables $X$ and $Y$ follow normal distributions with mean 0 and variances $\sigma^2$ and $\frac{\sigma^2}{4}$ respectively, and random variable $Z$ follows the standard normal distribution. For two positive numbers $a$ and $b$, $$\mathrm{P}(|X| \leqq a) = \mathrm{P}(|Y| \leqq b)$$ Which of the following statements in are correct? [4 points] ㄱ. $a > b$ ㄴ. $\mathrm{P}\left(Z > \frac{2b}{\sigma}\right) = \mathrm{P}\left(Y > \frac{a}{2}\right)$ ㄷ. If $\mathrm{P}(Y \leqq b) = 0.7$, then $\mathrm{P}(|X| \leqq a) = 0.3$.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

6. The measurement result of a certain physical quantity follows a normal distribution $N \left( 10 , \sigma ^ { 2 } \right)$. Which of the following conclusions is incorrect? ( )
A. The smaller $\sigma$ is, the greater the probability that the physical quantity falls in $( 9.9,10.1 )$ in a single measurement.
B. The smaller $\sigma$ is, the probability that the physical quantity is greater than 10 in a single measurement is 0.5.
C. The smaller $\sigma$ is, the probability that the physical quantity is less than 9.99 equals the probability that it is greater than 10.01 in a single measurement.
D. The smaller $\sigma$ is, the probability that the physical quantity falls in $( 9.9,10.2 )$ equals the probability that it falls in $( 10,10.3 )$ in a single measurement.
【Answer】D

【Solution】

【Analysis】Use the properties of the normal distribution density curve to judge each option. 【Detailed Solution】For option A, $\sigma ^ { 2 }$ is the variance of the data, so the smaller $\sigma$ is, the more concentrated the data is around $\mu = 10$. Therefore, the probability that the measurement result falls in $( 9.9,10.1 )$ increases, so A is correct.
For option B, by the symmetry of the normal distribution density curve, the probability that the physical quantity is greater than 10 in a single measurement is 0.5, so B is correct.
For option C, by the symmetry of the normal distribution density curve, the probability that the physical quantity is greater than 10.01 equals the probability that it is less than 9.99 in a single measurement, so C is correct.
For option D, since the probability that the physical quantity falls in $( 9.9,10.0 )$ does not equal the probability that it falls in $( 10.2,10.3 )$, the probability that the measurement result falls in $( 9.9,10.2 )$ does not equal the probability that it falls in $( 10,10.3 )$, so D is incorrect. Therefore, the answer is: D.

The measurement result of a certain physical quantity follows a normal distribution $N \left( 10 , \sigma ^ { 2 } \right)$. Which of the following conclusions is incorrect?
A. The smaller $\sigma$ is, the greater the probability that the physical quantity falls in $( 9.9,10.1 )$ in a single measurement.
B. The smaller $\sigma$ is, the probability that the physical quantity is greater than 10 in a single measurement is 0.5.
C. The smaller $\sigma$ is, the probability that the physical quantity is less than 9.99 equals the probability that it is greater than 10.01 in a single measurement.
D. The smaller $\sigma$ is, the probability that the physical quantity falls in $( 9.9,10.2 )$ equals the probability that it falls in $( 10,10.3 )$ in a single measurement.

To understand the per-acre income (in units of 10,000 yuan) after promoting exports, a sample was taken from the planting area. The sample mean of per-acre income after promoting exports is $\bar { x } = 2.1$ , and the sample variance is $s ^ { 2 } = 0.01$ . The historical per-acre income $X$ in the planting area follows a normal distribution $N \left( 1.8 , ~ 0.1 ^ { 2 } \right)$ . Assume that the per-acre income $Y$ after promoting exports follows a normal distribution $N \left( \bar { x } , s ^ { 2 } \right)$ . Then (if a random variable $Z$ follows a normal distribution $N \left( \mu , \sigma ^ { 2 } \right)$ , then $P ( Z < \mu + \sigma ) \approx 0.8413$ )
A. $P ( X > 2 ) > 0.2$
B. $P ( X > 2 ) < 0.5$
C. $P ( Y > 2 ) > 0.5$
D. $P ( Y > 2 ) < 0.8$

In general, the following inequality holds for a random variable $X$ with expected value $\mu$ and standard deviation $\sigma$ for $k > 0$ :
$$P ( \mu - k \cdot \sigma < X < \mu + k \cdot \sigma ) \geq 1 - \frac { 1 } { k ^ { 2 } }$$
Explain the statement of this inequality for $k = 2$.

10. The figure below is a histogram based on the weights of 100 women (the percentages in the figure represent the relative frequency of each weight interval, where each interval does not include the left endpoint but includes the right endpoint). The mean weight of the 100 women is 55 kg, and the standard deviation is 12.5 kg. Curve N represents a normal distribution with the same mean and standard deviation as the sample values. In this sample, if "overweight" is defined as weight exceeding the sample mean by 2 or more standard deviations (i.e., weight exceeding 80 kg or more), which of the following statements are correct? [Figure]
(1) In curve N (normal distribution), the proportion at 55 kg or above is approximately 50\%.
(2) In curve N (normal distribution), the proportion at 80 kg or above is approximately 2.5\%.
(3) In this sample, the median weight is greater than 55 kg.
(4) In this sample, the first quartile of weight is greater than 45 kg.
(5) In this sample, the proportion of "overweight" (weight exceeding 80 kg or more) is greater than or equal to 5\%.

9. A company commissioned a polling organization to survey the percentage of residents in locations A and B who have heard of a certain product (hereinafter referred to as ``awareness''). The results are as follows: At a 95\% confidence level, the confidence intervals for the product's awareness in locations A and B are $[0.50, 0.58]$ and $[0.08, 0.16]$ respectively. Which of the following options are correct?
(1) In location A, 54\% of the survey respondents have heard of the product
(2) The number of survey respondents in location B was less than in location A
(3) The survey results can be interpreted as: the probability that more than half of all residents in location A have heard of the product is greater than 95\%
(4) If multiple surveys are conducted in location B using the same method, the awareness has a 95\% chance of falling in the interval $[0.08, 0.16]$
(5) After intensive advertising, a follow-up survey is conducted in location B with the number of respondents increased to four times the original number. Then at a 95\% confidence level, the width of the confidence interval for the product's awareness will be reduced by half (i.e., 0.04)