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The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are:
- a)10, 11
- b)8, 13
- c)1, 20
- d)3, 18
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The mean of 6 distinct observations is 6.5 and their variance is 10.25...
Let the other two numbers be a, (21 - a).
Now,
10.25 =
(Using the formula for variance)
⇒ 6(10.25) + 6(6.5)2 = 94 + a2 + (21 - a)2
⇒ a2 + (21 - a2) = 221
a = 10 and (21 - a) = 21 - 10 = 11
So, the remaining two observations are 10 and 11.
Now,
10.25 =
(Using the formula for variance)
⇒ 6(10.25) + 6(6.5)2 = 94 + a2 + (21 - a)2
⇒ a2 + (21 - a2) = 221
a = 10 and (21 - a) = 21 - 10 = 11
So, the remaining two observations are 10 and 11.
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Community Answer
The mean of 6 distinct observations is 6.5 and their variance is 10.25...
Given information:
- Mean of 6 distinct observations = 6.5
- Variance of the observations = 10.25
To find the remaining two observations, we need to determine the sum of all 6 observations and the sum of their squares.
Let's denote the remaining two observations as x and y.
The sum of all 6 observations can be calculated as follows:
2 + 4 + 5 + 7 + x + y = 6.5 * 6
18 + x + y = 39
x + y = 21 ...........(1)
The sum of squares of the 6 observations can be calculated as follows:
2^2 + 4^2 + 5^2 + 7^2 + x^2 + y^2 = 10.25 * 6
4 + 16 + 25 + 49 + x^2 + y^2 = 61.5
x^2 + y^2 = 67.5 - 94
x^2 + y^2 = -26.5 ...........(2)
We have two equations (1) and (2) with two variables (x and y). We can solve these equations simultaneously to find the values of x and y.
Solving equation (1) for y, we get:
y = 21 - x
Substituting this value in equation (2), we have:
x^2 + (21 - x)^2 = -26.5
x^2 + 441 - 42x + x^2 = -26.5
2x^2 - 42x + 467.5 = 0
Dividing the equation by 2, we get:
x^2 - 21x + 233.75 = 0
Using the quadratic formula, we can find the values of x:
x = (21 ± √(21^2 - 4*1*233.75))/(2*1)
x = (21 ± √(441 - 935))/(2)
x = (21 ± √(-494))/(2)
Since we are looking for real values of x and y, the discriminant (b^2 - 4ac) must be greater than or equal to zero.
In this case, the discriminant is -494, which is negative. Therefore, there are no real solutions for x and y.
Hence, the given options do not provide the correct values for the remaining two observations.
Therefore, the correct answer is none of the given options.
- Mean of 6 distinct observations = 6.5
- Variance of the observations = 10.25
To find the remaining two observations, we need to determine the sum of all 6 observations and the sum of their squares.
Let's denote the remaining two observations as x and y.
The sum of all 6 observations can be calculated as follows:
2 + 4 + 5 + 7 + x + y = 6.5 * 6
18 + x + y = 39
x + y = 21 ...........(1)
The sum of squares of the 6 observations can be calculated as follows:
2^2 + 4^2 + 5^2 + 7^2 + x^2 + y^2 = 10.25 * 6
4 + 16 + 25 + 49 + x^2 + y^2 = 61.5
x^2 + y^2 = 67.5 - 94
x^2 + y^2 = -26.5 ...........(2)
We have two equations (1) and (2) with two variables (x and y). We can solve these equations simultaneously to find the values of x and y.
Solving equation (1) for y, we get:
y = 21 - x
Substituting this value in equation (2), we have:
x^2 + (21 - x)^2 = -26.5
x^2 + 441 - 42x + x^2 = -26.5
2x^2 - 42x + 467.5 = 0
Dividing the equation by 2, we get:
x^2 - 21x + 233.75 = 0
Using the quadratic formula, we can find the values of x:
x = (21 ± √(21^2 - 4*1*233.75))/(2*1)
x = (21 ± √(441 - 935))/(2)
x = (21 ± √(-494))/(2)
Since we are looking for real values of x and y, the discriminant (b^2 - 4ac) must be greater than or equal to zero.
In this case, the discriminant is -494, which is negative. Therefore, there are no real solutions for x and y.
Hence, the given options do not provide the correct values for the remaining two observations.
Therefore, the correct answer is none of the given options.
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The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are:a)10, 11b)8, 13c)1, 20d)3, 18Correct answer is option 'A'. Can you explain this answer?
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