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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
7 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
one year 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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