There are two datasets A and B. The mean and median of dataset A, which consists of 5 distinct numbers, are both 25. Dataset B consists of 7 numbers, where 5 of them match the data in A, and the remaining 2 are $x$ and $y$. Which of the following statements are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Let $\bar { X }$ be the sample mean obtained by randomly extracting a sample of size $n$ from a population with population standard deviation 14. When $\sigma ( \bar { X } ) = 2$, what is the value of $n$? [3 points]
(1) 9
(2) 16
(3) 25
(4) 36
(5) 49

A bag contains 10 balls labeled with the number 1, 20 balls labeled with the number 2, and 30 balls labeled with the number 3. A ball is drawn at random from the bag, the number on the ball is noted, and the ball is returned. This procedure is repeated 10 times, and let $Y$ be the sum of the 10 numbers observed. The following is the process of finding the mean $\mathrm { E } ( Y )$ and variance $\mathrm { V } ( Y )$ of the random variable $Y$.
Consider the 60 balls in the bag as a population. When a ball is drawn at random from this population, let $X$ be the random variable representing the number on the ball. The probability distribution of $X$, which is the probability distribution of the population, is shown in the following table.

$X$	1	2	3	Total
$\mathrm { P } ( X = x )$	$\frac { 1 } { 6 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 2 }$	1

Therefore, the population mean $m$ and population variance $\sigma ^ { 2 }$ are
$$m = \mathrm { E } ( X ) = \frac { 7 } { 3 } , \quad \sigma ^ { 2 } = \mathrm { V } ( X ) = \text { (가) }$$
When a sample of size 10 is randomly extracted from the population and the sample mean is $\bar { X }$,
$$\mathrm { E } ( \bar { X } ) = \frac { 7 } { 3 } , \quad \mathrm {~V} ( \bar { X } ) = \text { (나) }$$
If the number on the $n$-th ball drawn from the bag is $X _ { n }$, then
$$Y = \sum _ { n = 1 } ^ { 10 } X _ { n } = 10 \bar { X }$$
so
$$\mathrm { E } ( Y ) = \frac { 70 } { 3 } , \quad \mathrm {~V} ( Y ) = \text { (다) }$$
If the numbers that fit (가), (나), and (다) are $p , q , r$ respectively, what is the value of $p + q + r$? [4 points]
(1) $\frac { 31 } { 6 }$
(2) $\frac { 11 } { 2 }$
(3) $\frac { 35 } { 6 }$
(4) $\frac { 37 } { 6 }$
(5) $\frac { 13 } { 2 }$

A bag contains 10 balls labeled with the number 1, 20 balls labeled with the number 2, and 30 balls labeled with the number 3. One ball is randomly drawn from the bag, the number on the ball is confirmed, and then the ball is returned. This procedure is repeated 10 times, and the sum of the 10 numbers confirmed is the random variable $Y$. The following is the process of finding the mean $\mathrm { E } ( Y )$ and variance $\mathrm { V } ( Y )$ of the random variable $Y$.
Let the 60 balls in the bag be the population. When one ball is randomly drawn from this population, let the number on the ball be the random variable $X$. The probability distribution of $X$, that is, the probability distribution of the population, is shown in the following table.

$X$	1	2	3	Total
$\mathrm { P } ( X = x )$	$\frac { 1 } { 6 }$	$\frac { 1 } { 3 }$	$\frac { 1 } { 2 }$	1

Therefore, the population mean $m$ and population variance $\sigma ^ { 2 }$ are $$m = \mathrm { E } ( X ) = \frac { 7 } { 3 } , \quad \sigma ^ { 2 } = \mathrm { V } ( X ) = \text { (가) }$$
When a sample of size 10 is randomly extracted from the population and the sample mean is $\bar { X }$, $$\mathrm { E } ( \bar { X } ) = \frac { 7 } { 3 } , \quad \mathrm {~V} ( \bar { X } ) = \text { (나) }$$
Let the number on the $n$-th ball drawn from the bag be $X _ { n }$. Then $$Y = \sum _ { n = 1 } ^ { 10 } X _ { n } = 10 \bar { X }$$ so $$\mathrm { E } ( Y ) = \frac { 70 } { 3 } , \quad \mathrm {~V} ( Y ) = \text { (다) }$$
When the numbers that fit (가), (나), and (다) are $p , q , r$ respectively, what is the value of $p + q + r$? [4 points]
(1) $\frac { 31 } { 6 }$
(2) $\frac { 11 } { 2 }$
(3) $\frac { 35 } { 6 }$
(4) $\frac { 37 } { 6 }$
(5) $\frac { 13 } { 2 }$

2. Given a set of data $4,6,5,8,7,6$, then the mean of this set of data is $\_\_\_\_$ .

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

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

The stem-and-leaf plot of the average monthly temperatures (in $^\circ$C) in Chongqing in 2013 is shown below.
The median of this data set is
(A) 19
(B) 20
(C) 21.5
(D) 23

A certain car fills up its fuel tank every time it refuels. The table below records the situation at two consecutive refueling times. Note: ``Cumulative mileage'' refers to the total distance the car has traveled since leaving the factory. During this period, the average fuel consumption per 100 kilometers for this car is\n\n\n

\n\nRefueling Date	Refueling Amount (liters)	\begin{tabular}{ c }\nCumulative Mileage at
\nRefueling (kilometers)
\n

\n\hline\nMay 1, 2015 & 12 & 35000 \n\hline\nMay 15, 2015 & 48 & 35600 \n\hline\n\end{tabular}\n

To evaluate the planting effectiveness of a crop, $n$ plots of experimental land were selected. The per-acre yields (in kg) of these $n$ plots are $x_1, x_2, \cdots, x_n$ respectively. Among the following indicators, which can be used to evaluate the stability of this crop's per-acre yield?
A. Mean of $x_1, x_2, \cdots, x_n$
B. Median of $x_1, x_2, \cdots, x_n$
C. Maximum value of $x_1, x_2, \cdots, x_n$
D. Standard deviation of $x_1, x_2, \cdots, x_n$

A speech competition has 9 judges who each give an original score for a contestant. When determining the contestant's final score, the highest and lowest scores are removed from the 9 original scores, leaving 7 valid scores. Compared with the 9 original scores, the numerical characteristic that remains unchanged for the 7 valid scores is
A. median
B. mean
C. variance
D. range

13. A school will select one person from three candidates (A, B, C) to participate in the city-wide middle school boys' 1500-meter race. The mean and variance of their 10 recent training times (in seconds) are shown in the following table:

	A	B	C
Mean	280	280	290
Variance	20	16	16

Based on the data in the table, the school should select \_\_\_\_ to participate in the race.

13. China's high-speed rail development is rapid and technologically advanced. According to statistics, among high-speed trains stopping at a certain station, 10 trains have an on-time rate of 0.97, 20 trains have an on-time rate of 0.98, and 10 trains have an on-time rate of 0.99. The estimated value of the average on-time rate for all high-speed trains stopping at this station is $\_\_\_\_$ .

14. China's high-speed rail development is rapid and technologically advanced. According to statistics, among high-speed trains stopping at a certain station, 10 trains have an on-time rate of 0.97, 20 trains have an on-time rate of 0.98, and 10 trains have an on-time rate of 0.99. The estimated value of the average on-time rate for all trains stopping at this station is $\_\_\_\_$ .

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: $\sqrt { 74 } \approx 8.602$ .

For a sample of data $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$ with variance 0.01, the variance of the data $10 x _ { 1 } , 10 x _ { 2 } , \cdots , 10 x _ { n }$ is
A. 0.01
B. 0.1
C. 1
D. 10

In a sample of data, the frequencies of $1,2,3,4$ are $p _ { 1 } , p _ { 2 } , p _ { 3 } , p _ { 4 }$ respectively, and $\sum _ { i = 1 } ^ { 4 } p _ { i } = 1$ . Among the following four cases, the one with the largest standard deviation is
A. $p _ { 1 } = p _ { 4 } = 0.1 , p _ { 2 } = p _ { 3 } = 0.4$
B. $p _ { 1 } = p _ { 4 } = 0.4 , p _ { 2 } = p _ { 3 } = 0.1$
C. $p _ { 1 } = p _ { 4 } = 0.2 , p _ { 2 } = p _ { 3 } = 0.3$
D. $p _ { 1 } = p _ { 4 } = 0.3 , p _ { 2 } = p _ { 3 } = 0.2$

Given that the median of $a, b, 1, 2$ is 3 and the mean is 4, find $ab =$ $\_\_\_\_$

9. CD

Solution: Based on the formulas for calculating the mean, median, standard deviation, and range of sample data, options $C$ and $D$ are correct.

9. Which of the following statistics can measure the dispersion of the sample $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$? ( )
A. The standard deviation of the sample $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$
B. The median of the sample $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$
C. The range of the sample $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$
D. The mean of the sample $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$
【Answer】AC 【Solution】 【Analysis】Determine which of the given options measure data dispersion and which measure central tendency.
【Detailed Solution】By the definition of standard deviation, standard deviation measures data dispersion. By the definition of median, median measures central tendency. By the definition of range, range measures data dispersion. By the definition of mean, mean measures central tendency. Therefore, the answer is: AC. [Detailed Solution] From the given conditions, $f ( x ) = \left| e ^ { x } - 1 \right| = \left\{ \begin{array} { l } 1 - e ^ { x } , x < 0 \\ e ^ { x } - 1 , x \geq 0 \end{array} \right.$ , then $f ^ { \prime } ( x ) = \left\{ \begin{array} { l } - e ^ { x } , x < 0 \\ e ^ { x } , x > 0 \end{array} \right.$ , Thus point $A \left( x _ { 1 } , 1 - e ^ { x _ { 1 } } \right)$ and point $B \left( x _ { 2 } , e ^ { x _ { 2 } } - 1 \right)$ , $k _ { A M } = - e ^ { x _ { 1 } } , k _ { B N } = e ^ { x _ { 2 } }$ , Therefore $- e ^ { x _ { 1 } } \cdot e ^ { x _ { 2 } } = - 1 , x _ { 1 } + x _ { 2 } = 0$ , So $AM: y - 1 + e ^ { x _ { 1 } } = - e ^ { x _ { 1 } } \left( x - x _ { 1 } \right) , M \left( 0 , e ^ { x _ { 1 } } x _ { 1 } - e ^ { x _ { 1 } } + 1 \right)$ , Thus $| A M | = \sqrt { x _ { 1 } ^ { 2 } + \left( e ^ { x _ { 1 } } x _ { 1 } \right) ^ { 2 } } = \sqrt { 1 + e ^ { 2 x _ { 1 } } } \cdot \left| x _ { 1 } \right|$ , Similarly $| B N | = \sqrt { 1 + e ^ { 2 x _ { 2 } } } \cdot \left| x _ { 2 } \right|$ , Therefore $\frac { | A M | } { | B N | } = \frac { \sqrt { 1 + e ^ { 2 x _ { 1 } } } \cdot \left| x _ { 1 } \right| } { \sqrt { 1 + e ^ { 2 x _ { 2 } } } \cdot \left| x _ { 2 } \right| } = \sqrt { \frac { 1 + e ^ { 2 x _ { 1 } } } { 1 + e ^ { 2 x _ { 2 } } } } = \sqrt { \frac { 1 + e ^ { 2 x _ { 1 } } } { 1 + e ^ { - 2 x _ { 1 } } } } = e ^ { x _ { 1 } } \in ( 0,1 )$ . Thus the answer is: $( 0,1 )$ [Key Point Explanation] The key to solving this problem is to use the geometric meaning of the derivative to transform the condition $x _ { 1 } + x _ { 2 } = 0$ , and after eliminating one variable, the calculation yields the solution.

IV. Answer Questions: This section contains 6 questions totaling 70 points. Solutions should include written explanations, proofs, or calculation steps.

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

Which of the following statistics can measure the dispersion of the sample $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$?
A. The standard deviation of the sample $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$
B. The median of the sample $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$
C. The range of the sample $x _ { 1 } , x _ { 2 } , \cdots , x _ { n }$
D. (option D not fully provided in source)

A factory compares the treatment effects of two processes (Process A and Process B) on the elasticity of rubber products through 10 paired experiments. In each paired experiment, two rubber products of the same material are selected, one is randomly chosen to be treated with Process A and the other with Process B. The elasticity rates of the rubber products treated by Process A and Process B are recorded as $x _ { i } , y _ { i } ( i = 1,2 , \cdots 10 )$ respectively. The experimental results are as follows:

Experiment Number $i$	1	2	3	4	5	6	7	8	9	10
Elasticity Rate $x _ { i }$	545	533	551	522	575	544	541	568	596	548
Elasticity Rate $y _ { i }$	536	527	543	530	560	533	522	550	576	536

Let $z _ { i } = x _ { i } - y _ { i } ( i = 1,2 , \cdots , 10 )$. Let $\bar { z }$ denote the sample mean of $z _ { 1 } , z _ { 2 } , \cdots , z _ { 10 }$ and $s ^ { 2 }$ denote the sample variance.
(1) Find $\bar { z }$ and $s ^ { 2 }$.
(2) Determine whether the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B. (If $\bar { z } \geqslant 2 \sqrt { \frac { s ^ { 2 } } { 10 } }$, then it is considered that the elasticity rate of rubber products treated by Process A is significantly higher than that treated by Process B; otherwise, it is not considered to be significantly higher.)

An agricultural research department planted a new type of rice on 100 rice paddies of equal area and obtained the yield per mu (unit: kg) for each paddy, with partial data organized in the table below

Yield per mu	[900, 950)	[950, 1000)	[1000, 1050)	[1100, 1150)	[1150, 1200)
Frequency	6	12	18	24	10

Based on the data in the table, the correct conclusion is
A. The median yield per mu of the 100 paddies is less than 1050 kg
B. The proportion of paddies with yield per mu below 1100 kg among the 100 paddies exceeds $80 \%$
C. The range of yield per mu of the 100 paddies is between 200 kg and 300 kg
D. The mean yield per mu of the 100 paddies is between 900 kg and 1000 kg

The mean of the sample data 2, 8, 14, 16, 20 is ( )
A. 8
B. 9
C. 12
D. 18