A company manufactures women's general handball balls certified by the International Handball Federation. The weight of handball balls produced by this company follows a normal distribution with mean 350 g and standard deviation 16 g. The company determines that there is a problem in the production process if the average weight of 64 randomly selected handball balls is 346 g or less, or 355 g or more. Using the standard normal distribution table below, what is the probability that the company determines there is a problem in the production process? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
2.00	0.4772
2.25	0.4878
2.50	0.4938
2.75	0.4970

(1) 0.0290
(2) 0.0258
(3) 0.0184
(4) 0.0152
(5) 0.0092

A company produces women's general handball balls certified by the International Handball Federation. The weight of handball balls produced by this company follows a normal distribution with mean 350 g and standard deviation 16 g. The company determines that there is a problem in the production process if the average weight of 64 randomly selected handball balls is 346 g or less or 355 g or more. Using the standard normal distribution table on the right, what is the probability that the company determines there is a problem in the production process? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
2.00	0.4772
2.25	0.4878
2.50	0.4938
2.75	0.4970

(1) 0.0290
(2) 0.0258
(3) 0.0184
(4) 0.0152
(5) 0.0092

(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

The internal pressure strength of bottles produced at a certain factory follows a normal distribution $\mathrm { N } \left( m , \sigma ^ { 2 } \right)$, and bottles with internal pressure strength less than 40 are classified as defective. The process capability index $G$ for evaluating the factory's process capability is calculated as $$G = \frac { m - 40 } { 3 \sigma }$$ When $G = 0.8$, what is the probability that a randomly selected bottle is defective, using the standard normal distribution table on the right? [4 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
2.2	0.4861
2.3	0.4893
2.4	0.4918
2.5	0.4938

(1) 0.0139
(2) 0.0107
(3) 0.0082
(4) 0.0062
(5) 0.0038

The internal pressure strength of bottles produced at a certain factory follows a normal distribution $\mathrm { N } \left( m , \sigma ^ { 2 } \right)$, and bottles with internal pressure strength less than 40 are classified as defective. The process capability index $G$ for evaluating the process capability of this factory is calculated as $$G = \frac { m - 40 } { 3 \sigma }$$ When $G = 0.8$, what is the probability that a randomly selected bottle is defective, using the standard normal distribution table below? [4 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
2.2	0.4861
2.3	0.4893
2.4	0.4918
2.5	0.4938

(1) 0.0139
(2) 0.0107
(3) 0.0082
(4) 0.0062
(5) 0.0038

[Probability and Statistics] One method to determine whether a certain bone fossil belongs to animal A or animal B is to use the length of a specific part. The length of this part in animal A follows a normal distribution $\mathrm { N } \left( 10,0.4 ^ { 2 } \right)$, and the length of this part in animal B follows a normal distribution $\mathrm { N } \left( 12,0.6 ^ { 2 } \right)$. If the length of this part is less than $d$, it is judged to be a fossil of animal A, and if it is at least $d$, it is judged to be a fossil of animal B. Find the value of $d$ such that the probability of judging an animal A fossil as an animal A fossil equals the probability of judging an animal B fossil as an animal B fossil. (The unit of length is cm.) [4 points]
(1) 10.4
(2) 10.5
(3) 10.6
(4) 10.7
(5) 10.8

The distance from a customer's home to a traditional market follows a normal distribution with mean 1740 m and standard deviation 500 m. Among customers whose distance from home to market is 2000 m or more, 15\% use private vehicles to come to the market, and among customers whose distance is less than 2000 m, 5\% use private vehicles. When one customer who came to the market using a private vehicle is randomly selected, what is the probability that the distance from this customer's home to the market is less than 2000 m? (Here, when $Z$ is a random variable following the standard normal distribution, use $\mathrm { P } ( 0 \leqq Z \leqq 0.52 ) = 0.2$ for calculation.) [3 points]
(1) $\frac { 3 } { 8 }$
(2) $\frac { 7 } { 16 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 9 } { 16 }$
(5) $\frac { 5 } { 8 }$

The distance from home to a traditional market for customers using a certain traditional market follows a normal distribution with mean 1740 m and standard deviation 500 m. Among customers whose distance from home to the market is 2000 m or more, 15\% use personal vehicles to come to the market, and among customers whose distance is less than 2000 m, 5\% use personal vehicles. When one customer who came to the market using a personal vehicle is randomly selected, what is the probability that the distance from this customer's home to the market is less than 2000 m? (Note: When $Z$ is a random variable following the standard normal distribution, calculate using $\mathrm { P } ( 0 \leqq Z \leqq 0.52 ) = 0.2$.) [3 points]
(1) $\frac { 3 } { 8 }$
(2) $\frac { 7 } { 16 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 9 } { 16 }$
(5) $\frac { 5 } { 8 }$

The duration of one use of a public bicycle in a certain city follows a normal distribution with mean 60 minutes and standard deviation 10 minutes. When 25 uses of the public bicycle are randomly sampled and surveyed, what is the probability that the total duration of the 25 uses is 1450 minutes or more, using the standard normal distribution table on the right? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

(1) 0.8351
(2) 0.8413
(3) 0.9332
(4) 0.9772
(5) 0.9938

The daily production of employees at a certain company varies depending on their length of service. The daily production of an employee with $n$ months of service ( $1 \leqq n \leqq 100$ ) follows a normal distribution with mean $a n + 100$ ( $a$ is a constant) and standard deviation 12. When the probability that the daily production of an employee with 16 months of service is 84 or less is 0.0228, find the probability that the daily production of an employee with 36 months of service is at least 100 and at most 142 using the standard normal distribution table.

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

[3 points]
(1) 0.7745
(2) 0.8185
(3) 0.9104
(4) 0.9270
(5) 0.9710

We want to investigate the proportion of antibody possession for a specific disease among Korean adults. Let $p$ be the proportion of antibody possession in the population, and let $\hat { p }$ be the proportion of antibody possession in a sample of $n$ people randomly selected from the population. Find the minimum value of $n$ such that the probability that $| \hat { p } - p | \leq 0.16 \sqrt { \hat { p } ( 1 - \hat { p } ) }$ is at least 0.9544. (where $Z$ is a random variable following the standard normal distribution and $\mathrm { P } ( 0 \leq Z \leq 2 ) = 0.4772$.) [4 points]

The length $X$ of products manufactured at a certain factory follows a normal distribution with mean $m$ and standard deviation 4. When $\mathrm { P } ( m \leq X \leq a ) = 0.3413$, what is the probability that the sample mean of the lengths of 16 products randomly selected from this factory is at least $a - 2$, using the standard normal distribution table on the right? (where $a$ is a constant and the unit of length is cm) [4 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.0228
(2) 0.0668
(3) 0.0919
(4) 0.1359
(5) 0.1587

The test scores of all students at a certain school follow a normal distribution with mean 500 and standard deviation 25. When one student is randomly selected from this school, what is the probability that the student's test score is at least 475 and at most 550, using the standard normal distribution table below? [3 points]

$z$	$\mathrm{P}(0 \leq Z \leq z)$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

(1) 0.7745
(2) 0.8185
(3) 0.9104
(4) 0.9270
(5) 0.9710

A random variable $X$ follows a normal distribution $\mathrm { N } \left( m , \sigma ^ { 2 } \right)$ and satisfies the following conditions.
(a) $\mathrm { P } ( X \geq 64 ) = \mathrm { P } ( X \leq 56 )$
(b) $\mathrm { E } \left( X ^ { 2 } \right) = 3616$ What is the value of $\mathrm { P } ( X \leq 68 )$ obtained using the table on the right? [3 points]
(1) 0.9104
(2) 0.9332
(3) 0.9544
(4) 0.9772
(5) 0.9938

$x$	$\mathrm { P } ( m \leq X \leq x )$
$m + 1.5 \sigma$	0.4332
$m + 2 \sigma$	0.4772
$m + 2.5 \sigma$	0.4938

A pharmaceutical company produces medicine bottles with capacity following a normal distribution with mean $m$ and standard deviation 10. When a random sample of 25 bottles is taken from the company's production, the probability that the sample mean capacity is at least 2000 is 0.9772. Using the standard normal distribution table below, what is the value of $m$? (Here, the unit of capacity is mL.) [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.5	0.4332
2.0	0.4772
2.5	0.4938
3.0	0.4987

(1) 2003
(2) 2004
(3) 2005
(4) 2006
(5) 2007

A snack factory produces snacks where the weight of one package follows a normal distribution with mean 75 g and standard deviation 2 g. Using the standard normal distribution table below, what is the probability that the weight of a randomly selected package of snacks from this factory is at least 76 g and at most 78 g? [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.0440
(2) 0.0919
(3) 0.1359
(4) 0.1498
(5) 0.2417

A research institute investigated the length of tomato seedling stems 3 weeks after planting. The length of the tomato stems follows a normal distribution with mean 30 cm and standard deviation 2 cm. Using the standard normal distribution table on the right, find the probability that the length of a randomly selected tomato stem is at least 27 cm and at most 32 cm. [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

(1) 0.6826
(2) 0.7745
(3) 0.8185
(4) 0.9104
(5) 0.9270

At a rice collection event, the weight of rice donated by each student follows a normal distribution with mean 1.5 kg and standard deviation 0.2 kg. When one student is randomly selected from those who participated in the event, what is the probability that the weight of rice donated by this student is at least 1.3 kg and at most 1.8 kg, using the standard normal distribution table below? [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.00	0.3413
1.25	0.3944
1.50	0.4332
1.75	0.4599

(1) 0.8543
(2) 0.8012
(3) 0.7745
(4) 0.7357
(5) 0.6826

The random variable $X$ follows a normal distribution with mean $m$ and standard deviation 5, and the probability density function $f ( x )$ of the random variable $X$ satisfies the following conditions.
(a) $f ( 10 ) > f ( 20 )$
(b) $f ( 4 ) < f ( 22 )$ When $m$ is a natural number, what is the value of $\mathrm { P } ( 17 \leq X \leq 18 )$ obtained using the standard normal distribution table on the right? [4 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.6	0.226
0.8	0.288
1.0	0.341
1.2	0.385
1.4	0.419

(1) 0.044
(2) 0.053
(3) 0.062
(4) 0.078
(5) 0.097

The random variable $X$ follows a normal distribution with mean $m$ and standard deviation 5, and the probability density function $f ( x )$ of the random variable $X$ satisfies the following conditions. (가) $f ( 10 ) > f ( 20 )$ (나) $f ( 4 ) < f ( 22 )$ When $m$ is a natural number, $\mathrm { P } ( 17 \leq X \leq 18 ) = a$. Find the value of $1000a$ using the standard normal distribution table below. [4 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.6	0.226
0.8	0.288
1.0	0.341
1.2	0.385
1.4	0.419

The content volume of a cosmetic product produced by a factory follows a normal distribution with mean 201.5 g and standard deviation 1.8 g. Find the probability that the sample mean of 9 randomly selected cosmetic products from this factory is at least 200 g using the standard normal distribution table on the right. [4 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

(1) 0.7745
(2) 0.8413
(3) 0.9332
(4) 0.9772
(5) 0.9938

A random variable $X$ follows a normal distribution with mean $m$ and standard deviation $\sigma$, and $$\mathrm { P } ( X \leq 3 ) = \mathrm { P } ( 3 \leq X \leq 80 ) = 0.3$$ Find the value of $m + \sigma$. (Here, when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 0.25 ) = 0.1 , \mathrm { P } ( 0 \leq Z \leq 0.52 ) = 0.2$ for calculation purposes.) [4 points]

The commute time of employees at a certain company on a certain day follows a normal distribution with mean 66.4 minutes and standard deviation 15 minutes. Among employees whose commute time is 73 minutes or more, 40\% used the subway, and among employees whose commute time is less than 73 minutes, 20\% used the subway, with the remaining employees using other transportation. What is the probability that a randomly selected employee from those who commuted on that day used the subway? (Here, when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 0.44 ) = 0.17$.) [4 points]
(1) 0.306
(2) 0.296
(3) 0.286
(4) 0.276
(5) 0.266

The weight of one paprika harvested at a certain farm follows a normal distribution with mean 180 g and standard deviation 20 g. Using the standard normal distribution table below, what is the probability that the weight of one randomly selected paprika from this farm is at least 190 g and at most 210 g? [3 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.0440
(2) 0.0919
(3) 0.1359
(4) 0.1498
(5) 0.2417

The random variable $X$ follows the normal distribution $\mathrm { N } \left( 10,2 ^ { 2 } \right)$, and the random variable $Y$ follows the normal distribution $\mathrm { N } \left( m , 2 ^ { 2 } \right)$. The probability density functions of $X$ and $Y$ are $f ( x )$ and $g ( x )$ respectively.
$$f ( 12 ) \leq g ( 20 )$$
For $m$ satisfying this condition, what is the maximum value of $\mathrm { P } ( 21 \leq Y \leq 24 )$ using the standard normal distribution table on the right? [4 points]

$z$	$\mathrm { P } ( 0 \leq Z \leq z )$
0.5	0.1915
1.0	0.3413
1.5	0.4332
2.0	0.4772

(1) 0.5328
(2) 0.6247
(3) 0.7745
(4) 0.8185
(5) 0.9104