GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise

   

Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise

Question 1.
The mean and variance of 8 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.
Solution:
Let the remaining two observations be x and y. Then,
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 1
⇒ 74 = 210 + x2 + y2 – 216
⇒ 74 + 216 – 210 = x2 + y2 i.e., x2 + y2 = 80
or (x + y)2 – 2xy = x2 + y2 = 80
i.e; 122 – 2xy = 80
⇒ – 2xy = 80 – 144
⇒ – 2xy = – 64
So, (x – y)2 = x2 + y2 – 2xy
= 80 – 64 = 16
⇒ x – y = ± 4.
On solving equations x + y = 12 and x – y = ± 4, we get
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 2
Thus, the remaining terms are 4 and 8.

GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise

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.
Solution:
Let the required remaining observations be x and y. Then,
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 3
⇒ 112 = 236 + x2 + y2 – 16 × 4
⇒ 112 = x2 + y2 + 236 – 224
⇒ x2 + y2 = 100
i.e; (x + y)2 = x2 + y2 + 2xy
⇒ 142 = 100 + 2xy
i.e; 196 – 100 = 2xy
⇒ 2xy = 96
So, (x – y)2 = x2 + y2 – 2xy
= 100 – 96 = 4
i.e; x – y = ±2 ……………………. (2)
Solving (1) and (2) we get
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 4
⇒ x = 6 and y = 8.
So, x = 6 and y = 8.
i.e; 6 and 8 are the remaining observations.

GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise

Question 3.
The mean and standard deviation of 6 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.
Solution:
Let the observations be x1, x2, x3, x4, x5 and x6. So,
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 5
On multiplying each observation by 3, we get the new observations as 3x1, 3x2, 3x3, 3x4, 3x5 and 3x6. Now, their mean
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 6
Variance of new observations
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 7
= \(\frac{9}{1}\) × Variance of old observations
= 9 × 42
= 144.
∴ Standard deviation of new observations
= \(\sqrt{Variance}\) = \(\sqrt{144}\) = 12.

GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise

Question 4.
Given that \(\bar {x} \) is the mean and σ2 is the variance of n observations x1, x2, x3, ……………., xn. Prove that the mean and variance of the observations ax1, ax2, ………….. axn are a\(\bar {x} \) and a2σ2 respectively (a ≠ 0).
Solution:
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 8

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 was incorrect. Caluculate the correct mean and standard deviation in each of the following cases:
(i) If the wrong item is omitted.
(ii) If it is replaced by 12.
Solution:
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 9
(i) Since the wrong item which has been omitted, the correct mean \(\frac{\sum_{i=1}^{20} x_{i}-8}{19}\), because on omitting the wrong item, the total number of observations left are 19.
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 10
Since incorrect mean and variance are given as 10 and 4 respectively, so
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 11
= 106.105 – 102.212
= 3.893.
Correct S.D. = \(\sqrt{3.893}\) = 1.97

(ii) The wrong item which was 8 had been replaced by 12. As such, the correct mean
= GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 12
= \(\frac{200}{20}\) + \(\frac{1}{5}\)
= 10 + 0.2 = 10.2.
Since incorrect mean and variance are given as 10 and 4 respectively, so
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 13

GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise

Question 6.
The mean and standard deviation of marks obtained by 50 students of a class in the three subjects Mathematics, Physics and Chemistry are given below:
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 14
Which of the three subjects shows the highest variablility in marks and which shows the least?
Solution:
Subject Mathematics
Mean \(\bar {x} \) = 42, standard deviation σ = 12.
∴ Coefficient of variance = \(\frac{σ}{x}\) × 100
= \(\frac{12}{42}\) × 100 = 28.56.
Mean \(\bar {x} \) = 32, standard deviation σ = 15.
∴ Coefficient of variation = \(\frac{σ}{x}\) × 100
\(\frac{15}{32}\) × 100
= 46.875.
Subject chemistry
Mean \(\bar {x} \) = 40.9, standard deviation σ = 20.
∴ Coefficient of variation = \(\frac{σ}{x}\) × 100
= \(\frac{20}{40.9}\) × 100 = 48.9.
Highest variability of marks is in Chemistry and it is least in Mathematics.

GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise

Question 7.
The mean and standard deviation of a group of 100 observations were found to be 20 and 3 respecitvely. 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 were omitted.
Solution:
Mean \(\bar {x} \) = \(\frac{\Sigma x_{i}}{n}\), \(\bar {x} \) = 20
∴ Σxi = n × \(\bar {x} \) = 100 × 20 = 2000.
∴ Incorrect value of Σxi = 2000.
∴ Correct value of Σxi = 2000 – 21 – 21 – 18 = 1940
Correct value of mean = \(\frac{1940}{97}\) = 20.
GSEB Solutions Class 11 Maths Chapter 15 Statistics Miscellaneous Exercise img 15
Incorrect value of Σxi2 = 40900.
Correct value of Σxi2 = 40900 – (21)2 – (21)2 – (18)2
= 40900 – 441 – 441 – 324
= 39694
∴ Corrected σ2 = \(\frac{39694}{97}\) – (20)2
= 409.22 – 400 = 9.22.
Correct value of standard deviation
σ = \(\sqrt{9.22}\) = 3.04.
Thus, corrected mean = 20
and correct value of σ = 3.04.

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