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Miscellaneous Exercise - Chapter 15 Statistics class 11 ncert solutions Maths - SaraNextGen [2024]


Question 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.

Answer:

Let the remaining two observations be x and y.

Therefore, the observations are 6, 7, 10, 12, 12, 13, xy.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5153/chapter%2015_html_m10f82996.gif

From (1), we obtain

x2 + y2 + 2xy = 144 …(3)

From (2) and (3), we obtain

2xy = 64 … (4)

Subtracting (4) from (2), we obtain

x2 + y– 2xy = 80 – 64 = 16

⇒ x – y = ± 4 … (5)

Therefore, from (1) and (5), we obtain

x = 8 and y = 4, when x – y = 4

x = 4 and y = 8, when x – y = –4

Thus, the remaining observations are 4 and 8.

Question 2:

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

Answer:

Let the remaining two observations be x and y.

The observations are 2, 4, 10, 12, 14, xy.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5154/chapter%2015_html_m7877d721.gif

From (1), we obtain

x2 + y2 + 2xy = 196 … (3)

From (2) and (3), we obtain

2xy = 196 – 100

⇒ 2xy = 96 … (4)

Subtracting (4) from (2), we obtain

x2 + y– 2xy = 100 – 96

⇒ (x – y)2 = 4

⇒ x – y = ± 2 … (5)

Therefore, from (1) and (5), we obtain

x = 8 and y = 6 when x – y = 2

x = 6 and y = 8 when x – y = – 2

Thus, the remaining observations are 6 and 8.

Question 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.

Answer:

Let the observations be x1x2x3x4x5, and x6.

It is given that mean is 8 and standard deviation is 4.

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5155/chapter%2015_html_m79c74e04.gif

If each observation is multiplied by 3 and the resulting observations are yi, then

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5155/chapter%2015_html_m75bb0af4.gif

From (1) and (2), it can be observed that,

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5155/chapter%2015_html_2b0c9d4b.gif

Substituting the values of xi and https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5155/chapter%2015_html_m17fbb965.gif  in (2), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5155/chapter%2015_html_40679e4f.gif

Therefore, variance of new observations = https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5155/chapter%2015_html_m5917faea.gif

Hence, the standard deviation of new observations is https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5155/chapter%2015_html_m59ca55f0.gif

Question 4:

Given that https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5156/chapter%2015_html_m17fbb965.gif is the mean and σ2 is the variance of observations x1x2 … xn. Prove that the mean and variance of the observations ax1ax2ax3 …axare https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5156/chapter%2015_html_mffe580.gif  and a2 σ2, respectively (a ≠ 0).

Answer:

The given n observations are x1x2 … xn.

Mean = https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5156/chapter%2015_html_m17fbb965.gif

Variance = σ2

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5156/chapter%2015_html_47bcc289.gif

If each observation is multiplied by a and the new observations are yi, then

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5156/chapter%2015_html_1a936f5e.gif

Therefore, mean of the observations, ax1ax2 … axn, is https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5156/chapter%2015_html_m1bc75cf7.gif .

Substituting the values of xiand https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5156/chapter%2015_html_m17fbb965.gif in (1), we obtain

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5156/chapter%2015_html_3f9221c3.gif

Thus, the variance of the observations, ax1ax2 … axn, is a2 σ2.

Question 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.

Answer:

(i) Number of observations (n) = 20

Incorrect mean = 10

Incorrect standard deviation = 2

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5157/chapter%2015_html_40dad22.gif

That is, incorrect sum of observations = 200

Correct sum of observations = 200 – 8 = 192

∴ Correct mean https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5157/chapter%2015_html_d6c87bf.gif  https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5157/chapter%2015_html_m780a42db.gif

 

Standard deviation, σ = 1n∑i=1nxi2 – 1n2∑i=1nxi2⇒2 = 1n∑i=1nxi2 – 1n∑i=1nxi2

⇒2 = 1n∑i=1nxi2 – x¯2                      as, 1n∑i=1nx = x ⇒2 = 120×Incorrect∑i=1nxi2 – 102

⇒4 = 120×Incorrect∑i=1nxi2 – 100⇒120×Incorrect∑i=1nxi2 = 104

⇒Incorrect∑i=1nxi2 = 2080Now, correct∑i=1nxi2 = Incorrect∑i=1nxi2 – 82

⇒correct∑i=1nxi2 = 2080 – 64 = 2016

∴ correct Standard Deviation = 1ncorrect∑i=1nxi2 – correct mean2

⇒correct Standard Deviation = 119×2016 – 192192

⇒correct Standard Deviation = 201619-192192

⇒correct Standard Deviation = 144019 = 121019

⇒correct Standard Deviation = 12 × 3.16219 = 1.997(ii) When 8 is replaced by 12,

Incorrect sum of observations = 200

∴ Correct sum of observations = 200 – 8 + 12 = 204

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5157/chapter%2015_html_m97302dc.gif

Question 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:

Subject

Mathematics

Physics

Chemistry

Mean

42

32

40.9

Standard deviation

12

15

20

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

Answer:

Standard deviation of Mathematics = 12

Standard deviation of Physics = 15

Standard deviation of Chemistry = 20

The coefficient of variation (C.V.) is given by https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5158/chapter%2015_html_41759d49.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5158/chapter%2015_html_m647eaebe.gif

The subject with greater C.V. is more variable than others.

Therefore, the highest variability in marks is in Chemistry and the lowest variability in marks is in Mathematics.

Question 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 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.

Answer:

Number of observations (n) = 100

Incorrect mean https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5159/chapter%2015_html_5cb67bfb.gif

Incorrect standard deviation https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5159/chapter%2015_html_23b245ca.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5159/chapter%2015_html_3d36f5b5.gif

∴ Incorrect sum of observations = 2000

⇒ Correct sum of observations = 2000 – 21 – 21 – 18 = 2000 – 60 = 1940

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5159/chapter%2015_html_a928eb4.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/175/5159/chapter%2015_html_610749a2.gif

Also Read : Exercise-16.1-Chapter-16-Probability-class-11-ncert-solutions-Maths

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