the mean and sd for a b and 2 are 3 and 1 respectively the value of ab would be
Answers
So you have three values, which are : a, b, and 2.
The mean of the three values is 3.
So to determine that expression, add up the values and divide by 3 (the number of values):
(a + b + 2) / 3 = 3
simplify:
a + b + 2 = 9
a + b = 7
Now let's go to the standard deviation.
To solve this, you take the difference of each data point away from the mean, square them, average those values, then get the square root. we are told the result is 2√3, so we have:
(a - 3)² + (b - 3)² + (2 - 3)²
(a - 3)² + (b - 3)² + (-1)²
(a - 3)² + (b - 3)² + 1
Divide that by 3:
[(a - 3)² + (b - 3)² + 1] / 3
And get the square root:
√{[(a - 3)² + (b - 3)² + 1] / 3}
Let's rationalize that by multiplying both halves of the fraction by √3:
√{3[(a - 3)² + (b - 3)² + 1] / 9}
√{3[(a - 3)² + (b - 3)² + 1]} / 3
This value is equal to 2√3, so let's set them equal:
√{3[(a - 3)² + (b - 3)² + 1]} / 3 = 2√3
simplify, starting with multiplying both sides by 3:
√{3[(a - 3)² + (b - 3)² + 1]} = 6√3
Now squaring both sides:
3[(a - 3)² + (b - 3)² + 1] = 36 * 3
From here, let's divide both sides by 3:
(a - 3)² + (b - 3)² + 1 = 36
Subtract 1 from both sides:
(a - 3)² + (b - 3)² = 35
Now we can use the first equation and substitute an expression for a in terms of b, then solve for b:
a + b = 7
a = 7 - b
So we have:
(7 - b - 3)² + (b - 3)² = 35
(4 - b)² + (b - 3)² = 35
Square the binomials:
16 - 8b + b² + b² - 6b + 9 = 35
and simplify:
2b² - 14b + 25 = 35
2b² - 14b - 10 = 0
Divide both sides by 2:
b² - 7b - 5 = 0
Quadratic Formula:
b = [ -b ± √(b² - 4ac)] / (2a)
b = [ -(-7) ± √((-7)² - 4(1)(-5))] / (2 * 1)
b = [ 7 ± √(49 + 20)] / 2
b = [ 7 ± √(69)] / 2
So we have two values for b, so let's see what we get for a:
a = 7 - b
a = 7 - [ 7 - √(69)] / 2 and a = 7 - [ 7 + √(69)] / 2
a = 7 - [ 7/2 - √(69) / 2] and a = 7 - [ 7/2 + √(69) / 2]
a = 7 - 7/2 + √(69) / 2 and a = 7 - 7/2 - √(69) / 2
a = 14/2 - 7/2 + √(69) / 2 and a = 14/2 - 7/2 - √(69) / 2
a = 7/2 + √(69) / 2 and a = 7/2 - √(69) / 2
So a and b end up being the same values.
Before going on, as a test, let's solve for the mean and SD, using √69 approx 8.307, we get:
a = 7/2 + 8.307 / 2 and b = 7/2 - 8.307 / 2
a = 3.5 + 4.1535 and b = 3.5 - 4.1535
a = 7.6535 and b = -0.6535
Mean:
(7.6535 - 0.6535 + 2) / 3
9/3
3
Mean works out. Now SD:
(7.6535 - 3)² + (-0.6535 - 3)² + (2 - 3)²
4.6535² + (-3.6535)² + (-1)²
21.65506225 + 13.34806225 + 1
36.0031245
Divide that by 3:
36.0031245 / 3 = 12.0010415
Square root of that:
√12.0010415 = 3.46425
And 2√3 = 3.464101
Not exact, but since we rounded mid-way through, wouldn't be, but is close enough to be considered correct.
So now that we have values for a and b, we can solve for:
ab
[ 7 - √(69)] / 2 * [ 7 + √(69)] / 2
(7 - √69)(7 + √69) / 4
(49 + 7√69 - 7√69 - 69) / 4
(49 - 69) / 4
(-20) / 4
-5