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The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80 . Then which one of the following gives possible values a and b? i a = 1, b = 6 ii. a = 3, b = 4 iii. a = 0, b = 7 iv. a = 5, b = 2

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80 . Then which one of the following gives possible values a and b?
i a = 1, b = 6
ii. a = 3, b = 4
iii. a = 0, b = 7
iv. a = 5, b = 2

Grade:upto college level

2 Answers

Latika Leekha
askIITians Faculty 165 Points
6 years ago
The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80.
Now, using the deifnition of mean we have,
(a + b + 8 + 5 + 10)/ 5 = 6.
This gives a + b = 7. … (1)
Using the definition of variance, we get
[(a-6)2 + (b-6)2 + (8-6)2 + (5-6)2 + (10-6)2] / 5 = 6.8
Hence, we have (a2 + b2 + 36 + 36 – 12a -12b + 4 + 1 + 16) = 34.
(a2 + b2 - 12a -12b) = -59.
From (1), we can use b = 7 – a
Hence, the above equation reduces to
(a2 + (7-a)2 - 12a -12 (7-a)) = -59
Simplifying we get,
2a2 – 14a + 24 = 0
This gives a2 – 7a + 12 = 0
Hence, we have (a-3)(a-4) = 0
‘This yields a = 3 or 4.
If you want you can use these values of a to find the corresponding values of b, but since there is only one option that has the value of a as 3, hence, the correct option is (2).
Thanks & Regards
Latika Leekha
askITIians Faculty
Rishi Sharma
askIITians Faculty 646 Points
9 months ago
Dear Student,
Please find below the solution to your problem.

The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80.
Now, using the deifnition of mean we have,
(a + b + 8 + 5 + 10)/ 5 = 6.
This gives a + b = 7. … (1)
Using the definition of variance, we get
[(a-6)2 + (b-6)2 + (8-6)2 + (5-6)2 + (10-6)2] / 5 = 6.8
Hence, we have (a2 + b2 + 36 + 36 – 12a -12b + 4 + 1 + 16) = 34.
(a2 + b2 - 12a -12b) = -59.
From (1), we can use b = 7 – a
Hence, the above equation reduces to
(a2 + (7-a)2 - 12a -12 (7-a)) = -59
Simplifying we get,
2a2 – 14a + 24 = 0
This gives a2 – 7a + 12 = 0
Hence, we have (a-3)(a-4) = 0
‘This yields a = 3 or 4.
If you want you can use these values of a to find the corresponding values of b, but since there is only one option that has the value of a as 3, hence, the correct option is (2).

Thanks and Regards

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