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Page No 229:

Question 1:

Mark the correct alternative in each of the following:
Which one of the following is not a measure of central value?

(a) Mean

(b) Range

(c) Median

(d) Mode

Answer:

We know that mean, median and mode are all measures of central tendency.

Hence, the correct choice is (b).

Page No 229:

Question 2:

The mean of n observations is X. If k is added to each observation, then the new mean is

(a) X

(b) X + k

(c) X − k

(d) kX

Answer:

Let us take n observations.

Ifbe the mean of the n observations, then we have

Add a constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Hence, the correct choice is (b).

Page No 229:

Question 3:

The mean of n observations is X. If each observation is multiplied by k, the mean of new observations is

(a) kX¯

(b) X¯k

(c) X¯+k

(d) X¯-k

Answer:

Let us take n observations.

Ifbe the mean of the n observations, then we have

Multiply a constant k to each of the observations. Then the observations becomes

Ifbe the mean of the new observations, then we have

Hence, the correct choice is (a).

Page No 229:

Question 4:

The mean of a set of seven numbers is 81. If one of the numbers is discarded, the mean of the remaining numbers is 78. The value of discarded number is

(a) 98

(b) 99

(c) 100

(d) 101
 

Answer:

Given that the mean of 7 numbers is 81. Let us denote the numbers by.

Ifbe the mean of the n observations, then we have

Hence the sum of 7 numbers is

If one number is discarded then the mean becomes 78 and the total numbers becomes 6.

Let the number discarded is x.

After discarding one number the sum becomesand then the mean is

But it is given that after discarding one number the mean becomes 78.

Hence we have

Thus the excluded number is. So, the correct choice is (b).

Page No 229:

Question 5:

For which set of numbers do the mean, median and mode all have the same value?
(a) 2, 2, 2, 2, 4

(b) 1, 3, 3, 3, 5

(c) 1, 1, 2, 5, 6

(d) 1, 1, 1, 2, 5

Answer:

For the data 2, 2, 2, 2, 4 of 5 numbers, we have

Since, 2 occurs maximum number of times,

For the data 1, 3, 3, 3, 5 of 5 numbers, we have

Since, 3 occurs maximum number of times,

Hence, the correct choice is (b).

Note that if it happens that the result is not true for the second data then we must follow the same procedure for the other data’s.

Page No 229:

Question 6:

For the set of numbers 2, 2, 4, 5 and 12, which of the following statements is true?

(a) Mean = Median

(b) Mean > Mode

(c) Mean > Mode

(d) Mode = Median

Answer:

For the data 2, 2, 4, 5, 12 of 5 numbers, we have

Since, 2 occurs maximum number of times,

So, the correct choice is (b).

Page No 229:

Question 7:

If the arithmetic mean of 7, 5, 13, x and 9 is 10, then the value x is

(a) 10

(b) 12

(c) 14

(d) 16

Answer:

The given data is 7, 5, 13, x and 9. They are 5 in numbers.

The mean is

But, it is given that the mean is 10. Hence, we have

Hence, the correct choice is (d).

Page No 229:

Question 8:

Mode is

(a) least frequent value

(b) middle most value

(c) most frequent value

(d) none of these

Answer:

We know that, mode is the observation which occur maximum number of times.

Hence, the correct choice is (c).

Page No 229:

Question 9:

The following is the data of wages per day : 5, 4, 7, 5, 8, 8, 8, 5, 7, 9, 5, 7, 9, 10, 8
The mode of the data is

(a) 7

(b) 5

(c) 8

(d) 10

Answer:

The given data is 5, 4, 7, 5, 8, 8, 8, 5, 7, 9, 5, 7, 9, 10 and 8.

Make the following frequency table.

Since the values 5 and 8 occurs in the data maximum number of times, that is, 4. Hence, the modal value is 5 and 8. In this case the mode is not unique.

Hence, the correct options are (b) and (c).

Page No 229:

Question 10:

The median of the following data : 0, 2, 2, 2, -3, 5, –1, 5, 5, –3, 6, 6, 5, 6 is

(a) 0

(b) −1.5

(c) 2

(d) 3.5

Answer:

The given data is 0, 2, 2, 2, -3, 5, -1, 5, 5, -3, 6, 6, 5 and 6.

Arranging the given data in ascending order, we have

-3, -3, -1, 0, 2, 2, 2, 5, 5, 5, 5, 6, 6, 6

Here, the number of observation, which is an even number.

Hence, the median is

So, the correct choice is (d).



Page No 230:

Question 11:

A, B, C are three sets of values of x:

(a) A: 2, 3, 7, 1, 3, 2, 3
(b) 7, 5, 9, 12, 5, 3, 8
(c) 4, 4, 11, 7, 2, 3, 4

Which one of the following statements is correct?

(a) Mean of A = Mode of C

(b) Mean of C = Median of B

(c) Median of B = Mode of A

(d)Mean, Median and Mode of A are equal.
 

Answer:

For the data A: 2, 3, 7, 1, 3, 2, 3 of 7 numbers, we have

Arranging the data A in ascending order, we have

A: 1, 2, 2, 3, 3, 3, 7

Since, 3 occurs maximum number of times,

Hence, the correct choice is (d).

Note that if it happens that the result is not found in the first step then we must follow the same procedure for the other data’s.

Page No 230:

Question 12:

The median of the data : 4,4,5,7,6,7,7,12,3 is 

(a) 4
(b) 5
(c) 6
(d) 7

Answer:


Arranging the data in ascending order, we have

3, 4, 4, 5, 6, 7, 7, 7, 12

Here, the number of observations n = 9, which is odd

∴ Median = Value of 9+12th observation = Value of 102th observation = Value of 5th observation = 6

Thus, the median of given data is 6.

Hence, the correct answer is option (c).

Page No 230:

Question 13:

The median of the data : 78,56,22,34,45,54,39,68,54,84 is 

(a) 45
(b) 49.5
(c) 53.5
(d) 56

Answer:


Disclaimer: Options (d) in the question changed to match the answer.

The median of the data : 78,56,22,34,45,54,39,68,54,84 is 

(a) 45
(b) 49.5
(c) 53.5
(d) 54

Solution:

Arranging the data in ascending order, we have

22, 34, 39, 45, 54, 54, 56, 68, 78, 84

Here, the number of observations n = 10, which is even

∴ Median of the data

Value of 102th observation+Value of 102+1th observation2

Value of 5th observation+Value of 6th observation2

= 54+542

= 1082

= 54

Thus, the median of the given data is 54.

Hence, the correct answer is option (d).

Page No 230:

Question 14:

Mode of the data : 15,14,19,20,14,15,16,14,15,18,14,19,15,17,15 is 

(a) 14
(b) 15
(c) 16
(d) 17

Answer:


Arranging the given data in ascending order, we have

14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 17, 18, 19, 19, 20

Mode is the value of observation which occurs most frequently in a set of observations.

It can be seen that the value 15 occurs maximum number of times i.e. 5.

Thus, the mode of given data is 15.

Hence, the correct answer is option (b).

Page No 230:

Question 15:

If the mean of five observations x, x+2, x+4, x+6, x+8, is 11, then the mean of first three observations is

(a) 9

(b) 11

(c) 13

(d) none of these

Answer:

The given data is x, x + 2,  x + 4, x + 6 and x + 8. They are 5 in numbers.

The mean is

But, it is given that the mean is 11. Hence, we have

Then the first three observations are 7, 7 + 2, 7+4, that is, 7, 9, 11. Their mean is

Hence, the correct choice is (a).

Page No 230:

Question 16:

The algebraic sum of the deviations of a set of n values from their mean is

(a) 0

(b) n − 1

(c) n

(d) n + 1

Answer:

Ifbe the mean of the n observations, then we have

Letbe the mean of n values. So, we have

The sum of the deviations of n valuesfrom their meanis

Hence the correct choice is (a).

Page No 230:

Question 17:

The mean of five numbers is 30. If one number is excluded, their mean becomes 28. The excluded number is
(a) 28
(b) 30
(c) 35
(d) 38

Answer:

Let the five numbers be abcd, and e. We have,
a+b+c+d+e5=30a+b+c+d+e=150      .....1

Now,
a+b+c+d4=28a+b+c+d=112    .....2

From (1) and (2), we have
e=38

Thus, the excluded number is 38.

Hence, the correct answer is option (d).
 

Page No 230:

Question 18:

If the mean of the observations: x, x + 3, x + 5, x + 7, x + 10 is 9, the mean of last three observation is

(a) 313

(b) 323

(c) 343

(d) 353

Answer:

Given that, mean of the observations: x, x + 3, x + 5, x + 7, x + 10 is 9
x+x+3+x+5+x+7+x+105=95x+255=95x+25=455x=20x=4
Now, the last three numbers are 9, 11 and 14.
 Mean=9+11+143=343

Hence, the correct answer is option (c).

Page No 230:

Question 19:

There are 50 numbers. Each number is subtracted from 53 and the mean of the numbers so obtained is found to be -3.5. The mean of the given numbers is 

(a) 46.5
(b) 49.5
(c) 53.5
(d) 56.5

Answer:


Let the 50 numbers be x1, x2, x3,..., x50.

When each number is subtracted from 53, the resulting numbers are 53 − x1, 53 − x2, 53 − x3,..., 53 − x50.

It is given that, the mean of resulting numbers 53 − x1, 53 − x2, 53 − x3,..., 53 − x50 is −3.5.

We know

Mean = Sum of observationsNumber of observations

-3.5=53-x1+53-x2+53-x3+...+53-x5050

53+53+53+...+5350 times-x1+x2+x3+...+x5050=-3.5

53×5050-x1+x2+x3+...+x5050=-3.5

x1+x2+x3+...+x5050=53+3.5=56.5        .....(1)

Now,

Mean of 50 given numbers = x1+x2+x3+...+x5050

∴ Mean of 50 given numbers = 56.5               [From (1)]

Thus, the mean of the given numbers is 56.5.

Hence, the correct answer is option (d).

Page No 230:

Question 20:

The mean of 100 observations is 50. If one of the observations which was 50 is replaced by 150, the resulting will be 

(a) 50.5
(b) 51
(c) 51.5
(d) 52

Answer:


We know

Mean = Sum of observationsNumber of observations

⇒ Sum of observations = Mean × Number of observations

Mean of 100 observations = 50      (Given)

∴ Sum of 100 observations = 50 × 100 = 5000

If one of the observations which was 50 is replaced by 150, then

New sum of 100 observations = Sum of 100 observations − 50 + 150

⇒ New sum of 100 observations = 5000 − 50 + 150 = 5100

∴ New mean of 100 observations = New sum of 100 observations100=5100100=51

Thus, the new mean of 100 observations is 51.

Hence, the correct answer is option (b).

Page No 230:

Question 21:

If X¯ is the mean of x1,x2,.......,xn , then for a0, the mean of  ax1,ax2,......,axn,  x1a,x2a,...,xnn is 

(a) a+1aX¯(b) 12a+1aX¯(c) a+1nX¯n(d) a+1aX¯2n

Answer:


It is given that, the mean of x1, x2,..., xn is X¯.

X¯=x1+x2+...+xnn                   Mean=Sum of observationsNumber of observations
   
x1+x2+...+xn=nX¯        .....(1)

Now, ax1,ax2,...,axn,x1a,x2a,...,xna are 2n observations.

∴ Mean of these observations

=ax1+ax2+...+axn+x1a+x2a+...+xna2n

=ax1+x2+...+xn+1ax1+x2+...+xn2n

=a+1ax1+x2+...+xn2n

=a+1a×nX2n                [Using (1)]

=12a+1aX

Thus, the mean of ax1,ax2,...,axn,x1a,x2a,...,xna is 12a+1aX.

Hence, the correct answer is option (b).

Page No 230:

Question 22:

Let X¯ be the mean of x1,x2,...,xn, and Y¯ the mean of  y1,y2,...,yn, if  Z¯ is the mean of  x1,x2,...,xn, y1,y2,...,yn, then Z¯ is equal to 

(a) X¯ +Y¯(b) X¯+Y¯2(c) X¯+Y¯n(d) X¯+Y¯2n

Answer:


It is given that, the mean of x1, x2,..., xn is X¯.

X¯=x1+x2+...+xnn           Mean=Sum of observationsNumber of observations
   
x1+x2+...+xn=nX¯        .....(1)

Also, the mean of y1, y2,..., yn is Y¯.

Y¯=y1+y2+...+ynn

y1+y2+...+yn=nY        .....(2)

Now, x1,x2,...,xn,y1,y2,...,yn are 2n observations. The mean of these 2n observations is Z¯.

Z=x1+x2+...+xn+y1+y2+...+yn2n

Z=nX+nY2n                [Using (1) and (2)]

Z=nX+Y2n

Z=X+Y2

Hence, the correct answer is option (b).

Page No 230:

Question 23:

If X¯1,X¯2,...,X¯k are the means of  n group with n1, n2,...,nk number  of observations respectively, then the mean X¯ of all the groups taken together is govern by 

(a) i=1kni X¯i(b) 1n2i=1k ni X¯i(c) i=1k ni X¯ii=1k ni(d) i=1k ni X¯i2n

Answer:


We know

Mean = Sum of observationsNumber of observations

⇒ Sum of observations = Mean × Number of observations


Mean of n1 observations of first group = X1

∴ Sum of n1 observations of first group = n1X1

Mean of n2 observations of second group = X2

∴ Sum of n2 observations of second group = n2X2

.            .            .            .            .            .            .
.            .            .            .            .            .            .

Mean of nk observations of kth group = Xk

∴ Sum of nk observations of kth group = nkXk

Now,

Sum of all observations in the k groups 

= Sum of n1 observations of first group + Sum of n2 observations of second group + ... + Sum of nk observations of kth group

= n1X1+n2X2+...+nkXk

= i=1kniXi                      .....(1)

Total number of observations in the k groups  = n1+n2+...+nk=i=1kni           .....(2)

∴ Mean X of all the groups = Sum of all observations in the k groupsTotal number of observations in the k groups=i=1kniXii=1kni            [Using (1) and (2)]        

Hence, the correct answer is option (c).



Page No 231:

Question 24:

The empirical relation between mean, mode and median is

(a) Mode = 3 Median − 2 Mean

(b) Mode = 2 Median − 3 Mean

(c) Median = 3 Mode − 2 Mean

(d) Mean = 3 Median − 2 Mode

Answer:

The relation between mean, median and mode is

Hence, the correct option is (a).

Page No 231:

Question 25:

The mean of a, b, c, d and e is 28. If the mean of a, c, and e is 24, What is the mean of b and d?

(a) 31

(b) 32

(c) 33

(d) 34

Answer:

Given that the mean of a, b, c, d and e is 28. They are 5 in numbers.

Hence, we have

But, it is given that the mean of a, c and e is 24. Hence, we have

Then, we have

Hence, the mean of b and d is.

Hence, the correct choice is (d).

Page No 231:

Question 26:

The mean of 25 observations is 36. Out of these observations if the mean of first 13 observations is 32 and that the last 13 observations is 40, the 13th observations is 

(a) 23
(b) 36
(c) 38
(d) 40

Answer:

We know

Mean = Sum of observationsNumber of observations

⇒ Sum of observations = Mean × Number of observations

Mean of 25 observations = 36                  (Given)

∴ Sum of 25 observations = 36 × 25 = 900

Mean of first 13 observations = 32           (Given)

∴ Sum of first 13 observations = 32 × 13 = 416

Mean of last 13 observations = 40           (Given)

∴ Sum of last 13 observations = 40 × 13 = 520

Now,

13th observation

= Sum of first 13 observations + Sum of last 13 observations − Sum of 25 observations

= 416 + 520 − 900

= 36

Thus, the 13th observation is 36.

Hence, the correct answer is option (b).

Page No 231:

Question 27:

If the median of x, y, z, p, q and r(x > y > z > p > q > r) is m, then the median of 2q, 2p, 2z, and 2y is

(a) m2

(b) m

(c) 2m

(d) cannot be determined
 

Answer:

We have,
 median of x, y, z, p, q and z+p2=m             .....(1)
                                             
Now,
 Median of 2q, 2p, 2z, and 2y = 2z+2p2
                                               = 2m

Hence, the correct answer is option (c).
                                           

 

Page No 231:

Question 28:

If mean of the following data is 6, then which of the following can be the value of a?
 

xi: 2 4 6 8 10
fi: 1 2 a 2 1

(a) 4
(b) 5
(c) 8
(d) all the above

Answer:

Given that, mean of data = 6
Mean=ΣxifiΣfi6=2+8+6a+16+106+a36+6a=36+6a

Thus a can take any value.

Hence, the correct answer is option (d).

Page No 231:

Question 29:

If the mode of the following data is 8, then the median is
 

xi: 2 4 6 p 10
fi: 1 3 5 7 2

(a) 4
(b) 6
(c) 8
(d) 10

Answer:

Given that, Mode = 8
Since mode is the data which has the maximum frequency.
Now, p has a maximum frequency of 7.
 p=8

xi: 2 4 6 8 10
fi: 1 3 5 7 2

Mean = =xififi=2+12+30+56+201+3+5+7+2=12018

Mode = 3Median − 2Mean

Median=Mode+2Mean3=8+12093=8+12093=7.11

Disclaimer: None of the above option is correct.

Page No 231:

Question 30:

If the mean of the following data is 5, then a =
 

xi: 1 3 5 7 a
fi: 4 2 6 4 4

(a) 9
(b) 8
(c) 11
(d) 12

Answer:

Given that, mean of data = 5
Mean=ΣxifiΣfi5=4+6+30+28+4a20100=68+4a4a=32a=8

Hence, the correct answer is option (b).

Page No 231:

Question 31:

Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) ​Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If the mean of a, b and c is same as the mean of b, c and d. Then a = d.
Statement-2 (Reason): If the mean of 2, a, and b is 8, then the mean of a, b and 8 is 10.

Answer:

Statement-1
Mean of a, b and c =a+b+c3      
Mean of b, c and b+c+d3

Now, since both the means are same
a+b+c3=b+c+d3a+b+c=b+c+da=d
Thus, statement-1 is true.

Statement-2
Mean of  2, a, and = 8
2+a+b3=8a+b=22

Mean of  8, a, and = 10
8+a+b3=10a+b=22
Thus, statement-2 is true but it is not the correct explanation of statement-1

Hence, the correct answer is option (b)



 

Page No 231:

Question 32:

Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) ​Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If the mode of the following data is 4, then the median is 5.

xi 2 3 p 5 6
fi 1 2 7 6 6

Statement-2 (Reason): Mode = 2 Mean – 3 Median.

Answer:

Statement-1
Mode = 4
Since the mode is the maximum frequency data.
Therefore, = 4

Now 
Mean=ΣxifiΣfi=2+6+28+30+3622=10222=5111
We know,
Mode = 3 Median  2 Mean (Statement-1)

Median=2Mean+Mode3=2×5111+43=102+4433=14633

Thus, the above statement is false.


Statement-2:
False; Mode = 3 Median  2 Mean

Hence, the correct answer is option (c)



Page No 232:

Question 33:

Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) ​Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If for a moderately symmetric distribution: Mean = 28 and Median = 30, then Mode = 34.
Statement-2 (Reason): For a moderately symmetric distribution:  Mode = 3 Median – 2 Mean.

Answer:

Statement-1
Given that,  Mean = 28 and Median = 30,
Now,
Mode = 3×30-2×28
          = 90 - 56
          = 34
Thus, statement-1 is true.

Statement-2: Mode = 3 Median – 2 Mean (Empirical relationship)

So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Page No 232:

Question 34:

Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) ​Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If the median of a, b, c, d, e and f is k (a < b < c < d < e < f), then the median of b, c, d, e is also k.
Statement-2 (Reason): The median is not affected by the extreme values.

Answer:

Statement-1
Median of a, b, c, d, e and f  = c+d2=k
Median of b, c, d, e = c+d2=k

Thus, statement-1 is true.

Statement-2: True; The median is sensitive only to the value of the middle point or points; it is not sensitive to the values of all other points.

So, Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is option (a).

Page No 232:

Question 35:

Each of the following questions contains STATEMENT-1 (Assertion) and STATEMENT-2 (Reason) and has following four choices (a), (b), (c) and (d), only one of which is the correct answer. Mark the correct choice.
(a) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(c) Statement-1 is true, Statement-2 is false.
(d) ​Statement-1 is false, Statement-2 is true.
Statement-1 (Assertion): If the mean of the data 2, 8, 6, 5, 4, 5, 6, 3, 6, 4, 9, 1, 5, 6, 5 is given to be 5, then the mean of the data 6, 24, 18, 15, 12, 15, 18, 9, 18, 12, 27, 3, 15, 18, 15 is 15.
Statement-2 (Reason): If the mean of the data x1, x2, x3, ..., xn–1, xn is X, then the mean of the data ax1, ax2, ax3,..., axn-1, axn, is aX, a ≠ 0.

Answer:

Statement-1 
Given that, mean of the data 2, 8, 6, 5, 4, 5, 6, 3, 6, 4, 9, 1, 5, 6, 5 = 5
If the mean of the data x1x2x3, ..., xn–1xn is X, then the mean of the data ax1, ax2ax3,..., axn-1axn, is aX≠ 0. (Statement-2)
Therefore, mean of 6, 24, 18, 15, 12, 15, 18, 9, 18, 12, 27, 3, 15, 18, 15 = 5 × 3          (a = 3)
                                                                                                                   = 15
Thus, the above statement is true.

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Hence, the correct answer is an option (a).



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