Two dice $A$ and $B$ are rolled. Let the numbers obtained on $A$ and $B$ be $\alpha$ and $\beta$ respectively. If the variance of $\alpha - \beta$ is $\frac { p } { q }$, where $p$ and $q$ are co-prime, then the sum of the positive divisors of $p$ is equal to
(1) 72
(2) 36
(3) 48
(4) 31

If the mean and variance of five observations are $\frac { 24 } { 5 }$ and $\frac { 194 } { 25 }$ respectively and the mean of first four observations is $\frac { 7 } { 2 }$, then the variance of the first four observations is equal to
(1) $\frac { 4 } { 5 }$
(2) $\frac { 77 } { 12 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 105 } { 4 }$

Consider 10 observations $x_1, x_2, \ldots, x_{10}$, such that $\sum_{i=1}^{10} (x_i - \alpha) = 2$ and $\sum_{i=1}^{10} (x_i - \beta)^2 = 40$, where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. Then $\frac{\beta}{\alpha}$ is equal to:
(1) 2
(2) $\frac{3}{2}$
(3) $\frac{5}{2}$
(4) 1

Let $M$ denote the median of the following frequency distribution.

Class	$0 - 4$	$4 - 8$	$8 - 12$	$12 - 16$	$16 - 20$
Frequency	3	9	10	8	6

Then 20 M is equal to :
(1) 416
(2) 104
(3) 52
(4) 208

The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12 . The correct standard deviation is
(1) 1.8
(2) 1.94
(3) $\sqrt { 3.96 }$
(4) $\sqrt { 3.86 }$

If the variance of the frequency distribution

$x$	$c$	$2c$	$3c$	$4c$	$5c$	$6c$
$f$	2	1	1	1	1	1

is 160, then the value of $c \in N$ is
(1) 7
(2) 8
(3) 5
(4) 6

Let the mean and the variance of 6 observations $a, b, 68, 44, 48, 60$ be 55 and 194, respectively. If $a > b$, then $a + 3b$ is
(1) 200
(2) 190
(3) 180
(4) 210

Let $\mathrm { a } _ { 1 } , \mathrm { a } _ { 2 } , \ldots , \mathrm { a } _ { 10 }$ be 10 observations such that $\sum _ { \mathrm { k } = 1 } ^ { 10 } \mathrm { a } _ { \mathrm { k } } = 50$ and $\sum _ { \forall \mathrm { k } < \mathrm { j } } \mathrm { a } _ { \mathrm { k } } \cdot \mathrm { a } _ { \mathrm { j } } = 1100$. Then the standard deviation of $a _ { 1 } , a _ { 2 } , \ldots , a _ { 10 }$ is equal to:
(1) 5
(2) $\sqrt { 5 }$
(3) 10
(4) $\sqrt { 115 }$

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If $\mu$ and $\sigma ^ { 2 }$ denote the mean and variance of the correct observations respectively, then $15 \mu + \mu ^ { 2 } + \sigma ^ { 2 }$ is equal to $\_\_\_\_$ .

If the mean and variance of the data $65,68,58,44,48,45,60 , \alpha , \beta , 60$ where $\alpha > \beta$ are 56 and 66.2 respectively, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to

If the variance $\sigma^2$ of the data $$\begin{array}{cccccccc} x_i & 0 & 1 & 5 & 6 & 10 & 12 & 17 \\ f_i & 3 & 2 & 3 & 2 & 6 & 3 & 3 \end{array}$$ is $k$, then the value of $\lfloor k \rfloor$ is $\underline{\hspace{1cm}}$ (where $\lfloor \cdot \rfloor$ denotes the greatest integer function).

Let the mean and the standard deviation of the probability distribution

X	$\alpha$	1	0	- 3
$\mathrm { P } ( \mathrm { X } )$	$\frac { 1 } { 3 }$	K	$\frac { 1 } { 6 }$	$\frac { 1 } { 4 }$

be $\mu$ and $\sigma$, respectively. If $\sigma - \mu = 2$, then $\sigma + \mu$ is equal to $\_\_\_\_$

Let $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathrm { N }$ and $\mathrm { a } < \mathrm { b } < \mathrm { c }$. Let the mean, the mean deviation about the mean and the variance of the 5 observations $9,25 , \mathrm { a } , \mathrm { b } , \mathrm { c }$ be 18,4 and $\frac { 136 } { 5 }$, respectively. Then $2 \mathrm { a } + \mathrm { b } - \mathrm { c }$ is equal to $\_\_\_\_$

Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that $\sum_{i=1}^{10}(x_i - 2) = 30$, $\sum_{i=1}^{10}(x_i - \beta)^2 = 98$, $\beta > 2$, and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1 - 1) + 4\beta$, $2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta$, then $\frac{\beta\mu}{\sigma^2}$ is equal to:
(1) 100
(2) 120
(3) 110
(4) 90

Marks obtains by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is
(1) 52
(2) 48
(3) 44
(4) 40

For a statistical data $x_1, x_2, \ldots, x_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum_{i=1}^{10} x_i^2 = 371$. He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is
(1) 9
(2) 5
(3) 7
(4) 4

The variance of the numbers $8, 21, 34, 47, \ldots, 320$ is

When processing two-dimensional data, one method is to project the data vertically onto a certain line and use that line as a number line to understand the variance of the one-dimensional data formed by the projection points. For the set of two-dimensional data shown in the figure, which line in the following options would result in the smallest variance of the one-dimensional projected data?
(1) $y = 2 x$
(2) $y = - 2 x$
(3) $y = - x$
(4) $y = \frac { x } { 2 }$
(5) $y = - \frac { x } { 2 }$

Classes A and B each have 40 students taking a mathematics exam (total score 100 points). After the exam, classes A and B adjust their scores using $y_1 = 0.8x_1 + 20$ and $y_2 = 0.75x_2 + 25$ respectively, where $x_1, x_2$ represent the original exam scores of classes A and B, and $y_1, y_2$ represent the adjusted scores of classes A and B. The average adjusted scores for both classes are 60 points, with adjusted standard deviations of 16 and 15 points respectively. Select the correct options.
(1) Every student in class A has an adjusted score not lower than their original score
(2) The average original score of class A is higher than that of class B
(3) The standard deviation of original scores in class A is higher than that in class B
(4) If student A from class A has a higher adjusted score than student B from class B, then A's original score is higher than B's original score
(5) If the number of students in class A with adjusted scores below 60 (failing) is greater than the number in class B, then the number of students in class A with original scores below 60 must be greater than in class B

A school's midterm examination has 29 test takers, and all scores are different. After statistics, the scores at the 25th, 50th, 75th, and 95th percentiles are 41, 60, 74, and 92 points respectively. Later, it was discovered that the scores needed adjustment. The scores of the top 15 students with higher scores should each be increased by 5 points, while the remaining students' scores remain unchanged. Assuming the adjusted scores at the 25th, 50th, 75th, and 95th percentiles are $a$, $b$, $c$, and $d$ points respectively, which of the following options is the tuple $(a, b, c, d)$?
(1) $(41,60,74,92)$ (2) $(41,60,74,97)$ (3) $(41,65,79,97)$ (4) $(46,65,79,92)$ (5) $(46,65,79,97)$

A game has a total of 210 players, each holding gems. Among them, 1 player holds 1 gem, 2 players hold 2 gems, and so on, with 20 players holding 20 gems. What is the 90th percentile of the number of gems held by each player?
(1) 16
(2) 17
(3) 18
(4) 19
(5) 20

A city is divided into east and west regions. Each region has a temperature monitoring station. The city's daily maximum temperature (in degrees Celsius) is recorded as the maximum of the daily temperatures from the two regions. The table below shows the distribution of daily maximum temperatures for the east and west regions over a certain month (30 days).

Temperature $t$	$18 \leq t < 24$	$24 \leq t < 30$	$30 \leq t < 36$	$36 \leq t$
East Region (days)	0	11	14	5
West Region (days)	3	12	15	0

Based on the above table, the distribution of the city's daily maximum temperature for the month is shown in the following table.

Temperature $t$	$18 \leq t < 24$	$24 \leq t < 30$	$30 \leq t < 36$	$36 \leq t$
Days	$A$	$B$	$C$	$D$

Select the option that could be the tuple $(A , B , C , D)$.
(1) $( 0,15,15,0 )$
(2) $( 3,12,15,5 )$
(3) $( 0,9,16,5 )$
(4) $( 3,7,15,5 )$
(5) $( 0,12,13,5 )$

A badminton player competes against four opponents: A, B, C, and D, one match each. After the competition, data from these four matches were collected, recording the total number of smashes by each opponent and the average and standard deviation of the time used per smash. The results are shown in the table below. For example, opponent A made 25 smashes in that match, with an average time of 1.2 seconds per smash and a standard deviation of 0.5 seconds.

Opponent	Number of smashes in that match	Average time per smash (seconds)	Standard deviation of time per smash (seconds)
A	25	1.2	0.5
B	14	1.5	0.3
C	20	1.7	0.2
D	30	1.2	0.4

Based on the above, regarding the performance of opponents A, B, C, and D, select the correct options.
(1) C had the highest average time per smash among the four in that match
(2) D spent the most total time on smashing among the four in that match
(3) A's time per smash in that match was the same as D's for every smash
(4) The range of A's smash times in that match is greater than the range of D's smash times in that match
(5) It is impossible for all of B's smash times in that match to be between 1.4 and 1.6 seconds

In a foreign language course, the average age of students in classes $A , B$ and $C$ is 20, 26 and 29 respectively. The average age of students in classes A and B together is 23, and the average age of students in classes B and C together is 28.
According to this, what is the average age of all students in these three classes?
A) 25,5
B) 26
C) 26,5
D) 27
E) 27,5

The graph below provides some information about the heights of five people.
The following are known about these people.

• Ay\c{s}e and Kemal are the same height.
• Bora is 2 cm shorter than Kemal.
• Elif is 6 cm taller than Mehmet.
• Mehmet is 3 cm taller than Ay\c{s}e.

Accordingly, what is the average height of these people in cm?
A) 164
B) 165
C) 166
D) 167
E) 168