How to solve y for this problem
1 answer:
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Answer:
a) μ = 25,σ = 40 b) x₃ = 0.27864, x₄ = 89.72
Step-by-step explanation:
Given:
{x} = {x₁,x₂,x₃,x₄} -----------equation (1)
where x₁ = 25,x₂ = -15, x₃ ≤ x₄, ^x₁ = 0, ^x₂=-1
So, equation 1 becomes
{x} = {25,-15,x₃,x₄}
a)
Now, as ^x₁ = (x₁-μ)/σ
0 = (25-μ)/σ
μ = 25 = mean{x}
Similarly, ^x₂=(x₂-μ)/σ
-1=(-15-25)/σ
σ = 40 =std {x}
b)
As we know that (x₁+x₂+x₃+x₄)/4=μ
By putting values of x₁,x₂,μ we get,
x₃+x₄ = 90 -----------equation (2)
Also we know that
σ² = (1/4)∑₁⁴²-μ²
40² = 0.25*[25²+(-15)²+x₃²+x₄²]-25²
By solving it we get
x₃²+x₄² = 8050 ----------------equation (3)
Now using equation 2
x₃+x₄ = 90
x₃ = 90-x₄
taking square on both sides
x₃² = 8100 +x₄² -180x₄
bu putting this value in equation 3, we get
8100 + x₄² -180x₄ + x₄² = 8050
-90x₄ + x₄² + 25 = 0
By solving this quadratic equation, we get
x₄ = 89.72 or 0.27864
we will take x₄ = 89.72 as x₃+x₄ = 90 and x₃ ≤ x₄
putting it in equation 2, we get
x₃ = 0.27864