Question 165
O gráfico mostra a distribuição de alunos de uma escola por faixa etária.
[Figure]
Se a escola tem 800 alunos no total, o número de alunos com idade entre 15 e 17 anos é
(A) 160 (B) 200 (C) 240 (D) 280 (E) 320

A company recorded its performance in a given year through the graph, with monthly data of total sales and expenses.
Monthly profit is obtained by subtracting expenses from total sales, in that order.
Which three months of the year had the highest profits recorded?
(A) July, September, and December.
(B) July, September, and November.
(C) April, September, and November.
(D) January, September, and December.
(E) January, April, and June.

A bookstore sells books of the following literary genres: science fiction, self-help, romance, and biography. The graph presents the inventory of books that this bookstore has, by literary genre and by author's nationality, as well as the demand by literary genre, obtained through a survey conducted with its regular customers.
The bookstore manager will place an order for new copies only of the genre whose quantity in stock is insufficient to meet the demand identified by the survey.
The genre of book from which the manager should order more copies is
(A) science fiction, because it is the one with the highest demand.
(B) biography, because it is the genre with the lowest demand.
(C) self-help, because the quantity in stock is less than the demand.
(D) biography, because it is the genre with the smallest quantity of books in stock.
(E) romance, because it is the one with the smallest inventory of books by Brazilian authors.

A language school offers courses in English, Spanish, French, and German. The graphs present the percentage distribution of enrollments, by language, in 2023, and the distribution of the number of enrollments, by language, in 2024.
To plan the activities for 2025, the school manager estimated that the total number of enrollments will be the same as in 2024, and the percentage distribution of enrollments, by language, will be equal to that recorded in 2023.
According to this estimate, the number of enrollments in the French course for the year 2025 will be
(A) 2.
(B) 12.
(C) 20.
(D) 22.
(E) 40.

A survey was conducted on 200 students at a school regarding their preferences for experiential activities. The students who participated in this survey chose one of cultural experience or ecological research, and the number of students who chose each activity is as follows.

Classification	Cultural Experience	Ecological Research	Total
Male Students	40	60	100
Female Students	50	50	100
Total	90	110	200

When one student is randomly selected from the 200 students who participated in this survey and is a student who chose ecological research, what is the probability that this student is a female student? [3 points]
(1) $\frac { 5 } { 11 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 6 } { 11 }$
(4) $\frac { 5 } { 9 }$
(5) $\frac { 3 } { 5 }$

16. The following table shows the top 5 industries in job applications and recruitment for the first quarter of 2004 in a certain region:

Industry	Computer	Machinery	Marketing	Logistics	Trade
Job Applications	215830	200250	154676	74570	65280

Industry	Computer	Marketing	Machinery	Construction	Chemical
Job Openings	124620	102935	89115	76516	70436

If the employment situation of an industry is measured by the ratio of job applications to job openings in that industry, then based on the data in the table, the employment situation is definitely
A. Better in the computer industry than in the chemical industry.
B. Better in the construction industry than in the logistics industry.
C. Most tight in the machinery industry.
D. More tight in the marketing industry than in the trade industry.
III. Solution Questions (Total Score: 86 points)

10. A research group conducted a survey on urban air quality and divided 24 cities into three groups (A, B, C) by region, with corresponding numbers of cities being $4, 12, 8$. If stratified sampling is used to select 6 cities, then the number of cities to be selected from group C is $\_\_\_\_$

2. In a marathon competition, the stem-and-leaf plot in Figure 1 shows the results (in minutes) of 35 athletes. [Figure]
If the athletes are numbered 1-35 according to their results from best to worst, and 7 people are selected using systematic sampling, then the number of athletes with results in the interval [139, 151] is
A. 3
B. 4
C. 5
D. 6

2. A middle school has 110 teachers in the junior high division and 150 teachers in the senior high division. Their gender ratios are shown in the figure. The number of female teachers in the school is
A. 167
B. 137
C. 123
D. 93 [Figure]

3. The stem-and-leaf plot below shows the average monthly temperatures (${ } ^ { \circ } C$) in Chongqing in 2013:
The median of this data set is
A. $19$
B. $20$
C. $21.5$
D. $23$
4. ``$\mathrm { x } > 1$'' is ``$\log _ { \frac { 1 } { 2 } } ( \mathrm { x } + 2 ) < 0$'' a
A. necessary and sufficient condition
B. sufficient but not necessary condition
C. necessary but not sufficient condition
D. neither sufficient nor necessary condition

3. Based on the bar chart of China's annual carbon dioxide emissions (in units of 10,000 tons) from 2004 to 2013 shown below, which of the following conclusions is incorrect? [Figure]
A. Year-on-year comparison shows that 2008 had the most significant effect in reducing carbon dioxide emissions
B. China's efforts to control carbon dioxide emissions showed results in 2007
C. Since 2006, China's annual carbon dioxide emissions have shown a decreasing trend
D. Since 2006, China's annual carbon dioxide emissions are positively correlated with the year

Based on the bar chart of China's sulfur dioxide emissions (in units of 10,000 tons) from 2004 to 2013 shown below, which of the following conclusions is incorrect?
(A) Year-on-year comparison shows that 2008 had the most significant reduction in sulfur dioxide emissions
(B) China's treatment of sulfur dioxide emissions became evident in 2007
(C) Since 2006, China's annual sulfur dioxide emissions have shown a decreasing trend
(D) Since 2006, China's annual sulfur dioxide emissions are positively correlated with the year

The number of senior, middle-aged, and young teachers at a certain school is shown in the table below. Using stratified sampling to investigate the physical condition of teachers, in the sample drawn, there are 320 young teachers. Then the number of senior teachers in the sample is\n\n\n

\n\nCategory	Number of People
\n\nSenior Teachers	900
\n\nMiddle-aged Teachers	1800
\n\nYoung Teachers	1600
\n\nTotal	4300
\n\n

\n

12. In a marathon competition, the stem-and-leaf plot of the times (in minutes) of 35 athletes is shown in Figure 4 [Figure]
If the athletes are numbered 1-35 according to their times from best to worst, and 7 people are selected using systematic sampling, then the number of athletes with times in the interval $[139, 151]$ is $\_\_\_\_$.

267 students in the third year participated in the final examination. The ranking of Chinese, mathematics, and total scores of 37 students in a certain class in the entire grade is shown below. A, B, and C are three students in this class.\n\nBased on this examination score,\n(1) Between students A and B, the student whose Chinese score ranking is ahead of their total score ranking is\n(2) In the two subjects of Chinese and mathematics, the subject in which both students' score rankings are more advanced is

14. An e-commerce company conducted a statistical survey of 10,000 online shoppers' consumption in 2014. The consumption amount (in units of 10,000 yuan) falls within the interval [0.3, 0.9]. The frequency distribution histogram is shown in the figure below.
(1) In the histogram, $\mathrm { a } =$ $\_\_\_\_$.
(2) Among these shoppers, the number of shoppers with consumption amount in the interval [0.5, 0.9] is $\_\_\_\_$. [Figure]

17. An enterprise wants to understand the service quality of a certain department to its employees. It randomly surveyed 50 employees. Based on the evaluation scores of these 50 employees for the department, a frequency distribution histogram was drawn (as shown in the figure). The sample data are grouped into intervals [40, 50], [50, 60], [60, 70], [70, 80], [80, 90], [90, 100].
(1) Find the value of $a$ in the frequency distribution histogram;
(2) Estimate the probability that an employee's evaluation score is not less than 80;
(3) From the surveyed employees whose scores are in $[ 40,60 ]$, randomly select 2 people. Find the probability that both of their scores are in $[ 40,50 ]$. [Figure]

A supermarket randomly selected 1000 customers and recorded their purchasing of four products: A, B, C, and D. The data is organized in the table below, where ``✓'' indicates purchase and ``×'' indicates no purchase.\n\n\n

\n\n\backslashbox{Number of Customers}{Product}	A	B	C	D
\n\n100	✓	×	✓	✓
\n\n217	×	✓	×	✓
\n\n200	✓	✓	✓	×
\n\n300	✓	×	✓	×
\n\n85	✓	×	×	×
\n\n98	×	✓	×	×
\n\n

\n\n\n(I) Estimate the probability that a customer purchases both B and C\n(II) Estimate the probability that a customer purchases exactly 3 of the four products A, B, C, and D\n(III) If a customer has purchased product A, which of products B, C, and D is the customer most likely to have purchased?

(12 points)
To monitor the production process of a production line for a certain component, an inspector randomly selects one component every 30 minutes and measures its size (in cm). Below are the sizes of 16 components randomly selected by the inspector in one day:

Sampling Order	1	2	3	4	5	6	7	8
Component Size	9.95	10.12	9.96	9.96	10.01	9.92	9.98	10.04
Sampling Order	9	10	11	12	13	14	15	16
Component Size	10.26	9.91	10.13	10.02	9.22	10.04	10.05	9.95

$\sqrt{\sum_{i=1}^{16}(i - 8.5)^2} \approx 18.439$, $\sum_{i=1}^{16}(x_i - \bar{x})(i - 8.5) = -2.78$, where $x_i$ is the size of the $i$-th component sampled, $i = 1, 2, \cdots, 16$.
(1) Find the correlation coefficient $r$ of $(x_i, i)$ $(i = 1, 2, \cdots, 16)$, and determine whether it can be concluded that the size of components produced on this day does not systematically increase or decrease as the production process progresses.
(2) Among the components sampled in one day, if a component with size outside $(\bar{x} - 3s, \bar{x} + 3s)$ appears, it is considered that the production line may have experienced an abnormal situation on this day, and the production process needs to be checked.
(i) Based on the sampling results of this day, is it necessary to check the production process?
(ii) Data outside $(\bar{x} - 3s, \bar{x} + 3s)$ are called outliers. Remove the outliers and estimate the mean and standard deviation of the component sizes produced by this production line on this day. (Round to 0.01)
$$\text{Attachment: For a sample }(x_i, y_i) (i = 1, 2, \cdots, n), \text{ the correlation coefficient is } r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}}.$$
$$\sqrt{0.008} \approx 0.09.$$

19. (12 points) To understand the production situation of small and medium enterprises in the industry, a government department randomly surveyed 100 enterprises and obtained a frequency distribution table for the growth rate $y$ of production value in the first quarter compared to the previous year's first quarter.

Interval for $y$	$[ - 0.20,0 )$	$[ 0,0.20 )$	$[ 0.20,0.40 )$	$[ 0.40,0.60 )$	$[ 0.60,0.80 )$
Number of enterprises	2	24	53	14	7

(1) Estimate the proportion of enterprises with production value growth rate not less than $40\%$ and the proportion of enterprises with negative growth, respectively;
(2) Find the estimated values of the mean and standard deviation of the production value growth rate for this type of enterprise (use the midpoint of each interval as the representative value for data in that interval). (Accurate to 0.01)
Attachment: $\sqrt { 74 } \approx 8.602$ .

A student interest group randomly surveyed the air quality level and the number of people exercising in a certain park each day over 100 days in a city. The data is organized in the following table (unit: days):

Air Quality Level	$[ 0,200 ]$	$( 200,400 ]$	$( 400,600 ]$
1 (Excellent)	2	16	25
2 (Good)	5	10	12
3 (Slight Pollution)	6	7	8
4 (Moderate Pollution)	7	2	0

(1) Estimate the probability that the air quality level on a given day in the city is 1, 2, 3, or 4 respectively;
(2) Find the estimated average number of people exercising in the park on a given day (use the midpoint of each interval as the representative value for data in that interval);
(3) If the air quality level on a given day is 1 or 2, the day is called ``good air quality''; if the air quality level is 3 or 4, the day is called ``poor air quality''. Based on the given data, complete the following $2 \times 2$ contingency table and determine whether there is 95\% confidence to conclude that the number of people exercising in the park on a given day is related to the air quality of the city on that day.

	Number of people $\leqslant 400$	Number of people $> 400$
Good air quality
Poor air quality

Attachment: $K ^ { 2 } = \frac { n ( a d - b c ) ^ { 2 } } { ( a + b ) ( c + d ) ( a + c ) ( b + d ) }$,

$P \left( K ^ { 2 } \geqslant k \right)$	0.050	0.010	0.001
$k$	3.841	6.635	10.828

.

2. To understand the rural economic situation in a certain area, a sample survey was conducted on the annual household income of farmers in that area. The survey data on farmers' annual household income was organized into the following frequency distribution histogram: [Figure]
Based on this frequency distribution histogram, which of the following conclusions is incorrect?
A. The estimated proportion of farmers with annual household income below 4.5 ten thousand yuan is $6 \%$
B. The estimated proportion of farmers with annual household income not less than 10.5 ten thousand yuan is $10 \%$
C. The estimated average annual household income of farmers in this area does not exceed 6.5 ten thousand yuan
D. It is estimated that more than half of the farmers in this area have annual household income between 4.5 and 8.5 ten thousand yuan

2. To understand the rural economic situation in a certain area, a sampling survey was conducted on the annual household income of farmers. The survey data on farmers' annual household income was organized into the following frequency distribution histogram. Based on this frequency distribution histogram, which of the following conclusions is incorrect?
A. The estimated proportion of farmers with annual household income below 4.5 ten thousand yuan is 60\%
B. The estimated proportion of farmers with annual household income not less than 10.5 ten thousand yuan is 10\% [Figure]
C. The estimated average annual household income of farmers in this area does not exceed 6.5 ten thousand yuan
D. It is estimated that more than half of the farmers have annual household income between 4.5 and 8.5 ten thousand yuan

17. Two machine tools, Machine A and Machine B, produce the same type of product. Products are classified by quality into first-grade and second-grade products. To compare the quality of products from the two machines, 200 products were produced by each machine. The quality statistics are shown in the table below:

	First-grade	Second-grade	Total
Machine A	150	50	200
Machine B	120	80	200
Total	270	130	400

(1) What are the frequencies of first-grade products produced by Machine A and Machine B, respectively?
(2) Can we conclude with 99\% confidence that there is a difference in product quality between Machine A and Machine B? Attachment: $\mathrm { K } ^ { 2 } = \frac { n ( a d - b c ) ^ { 2 } } { ( a + b ) ( c + d ) ( a + c ) ( b + d ) }$,

$\mathrm { P } \left( \mathrm { K } ^ { 2 } \geqslant k \right)$	0.050	0.010	0.001
$k$	3.841	6.635	10.828

The weekly extracurricular sports time (in hours) for two students, A and B, over 16 weeks is shown in the stem-and-leaf plot below:

\multicolumn{1}{c|}{A}		\multicolumn{1}{|c}{B}
6	1	5.
8530	6.	3
7532	7.	46
6421	8.	12256666
	42	9.	0238
		10.	1

Which of the following conclusions is incorrect?
A. The sample median of A's weekly extracurricular sports time is 7.4
B. The sample mean of B's weekly extracurricular sports time is greater than 8
C. The estimated probability that A's weekly extracurricular sports time exceeds 8 hours is greater than 0.4
D. The estimated probability that B's weekly extracurricular sports time exceeds 8 hours is greater than 0.6