NCERT Solutions class-11 Maths Miscellaneous

Miscellaneous Exercise

1. The mean and variance of eight observations are 9 and 9.25 respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.

Ans. Let two required observations be and Then,

According to question,

……….(i)

Also

……….(ii)

When then on solving and we get and

When then on solving and we get and

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2. The mean and variance of 7 observations are 8 and 16 respectively. If five of the observations are 2, 4, 10, 12, 14, find the remaining two observations.

Ans. Let two required observations be and Then,

According to question,

……….(i)

Also

……….(ii)

When then on solving and we get and

When then on solving and we get and

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3. The mean and standard deviation of six observations are 8 and 4 respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.

Ans. Let six observations be then

According to questions,

Now, if each observation is multiplied by 3, then

New mean = = 24

Also

Now, if each observation is multiplied by 3, then

New Variance =

= 720 – 576 = 144

Therefore, New S.D. = = 12

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4. Given that is the mean and is the variance of observations Prove that the mean and variance of the observations are and respectively

Ans. Given:

Also

New Mean =

Also

New Variance =

=

=

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5. The mean and standard deviation of 20 observations are found to be 10 and 2 respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:

(i) if wrong item is omitted

(ii) if it is replaced by 12.

Ans. Given: and

Now,

Incorrect

Now

(i) If wrong item is omitted: then we have 19 observations.

Correct = Incorrect

Correct = 200 – 8 = 192

Correct Mean = = 10.1

Also Correct Incorrect

Correct Variance = (Correct ) – (Correct mean)²

= = =

Correct S.D. = = 1.997

(ii) If it is replaced by 12: then

Correct = Incorrect

Correct = 200 – 8 + 12 = 204

Correct Mean = = 10.2

Also Correct Incorrect

Correct Variance = (Correct )

= = =

Correct S.D. = = 11.989

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6. The mean and standard deviation of marks obtained by 50 students of a class in three subjects, Mathematics, Physics and Chemistry are given below:

Which of the three subjects shows the highest variability in marks and which shows the lowest?

Ans. For Mathematics, = 42 and = 12

C.V. of Mathematics = = 28.57%

For Physics, = 32 and = 15

C.V. of Physics = = 46.88%

For Chemistry, = 40.9 and = 20

C.V. of Chemistry = = 48.9%

Therefore, Chemistry with highest C.V. shows highest variability and Mathematics with lowest C.V. shows lowest variability.

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7. The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respectively. Later on it was found that three observations were incorrect, which were recorded as 12, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.

Ans. Given: and

Now,

Incorrect

Now

When wrong items 21, 21 and 18 are omitted from the data, then there will be 97 observations.

Correct = Incorrect

Correct = 2000 – 21 – 21 – 18 = 1940

Correct Mean = = 20

Also Correct Incorrect

Correct Variance = (Correct ) – (Correct mean)²

= = =

Correct S.D. = = 3.036