The population mean:
μ = 1 + 3 + 4 3 = 8 3 . \mu=\cfrac{1+3+4}{3}=\cfrac{8}{3}. μ = 3 1 + 3 + 4 = 3 8 .
The population variance:
σ 2 = ∑ ( x i − μ ) 2 ⋅ P ( x i ) , \sigma^2=\sum(x_i-\mu)^2\cdot P(x_i), σ 2 = ∑ ( x i − μ ) 2 ⋅ P ( x i ) ,
X − μ = { 1 − 8 3 , 3 − 8 3 , 4 − 8 3 } = X-\mu=\begin{Bmatrix}
1-\cfrac{8}{3},3-\cfrac{8}{3},4-\cfrac{8}{3}
\end{Bmatrix}= X − μ = { 1 − 3 8 , 3 − 3 8 , 4 − 3 8 } =
= { − 5 3 , 1 3 , 4 3 } , =\begin{Bmatrix}
-\cfrac{5}{3}, \cfrac{1}{3},\cfrac{4}{3}
\end{Bmatrix}, = { − 3 5 , 3 1 , 3 4 } ,
σ 2 = ( − 5 3 ) 2 ⋅ 1 3 + ( 1 3 ) 2 ⋅ 1 3 + ( 4 3 ) 2 ⋅ 1 3 = = 42 27 = 14 9 = 1.556. \sigma^2=\bigg(\cfrac{-5}{3}\bigg)^2\cdot \cfrac{1}{3}+\bigg(\cfrac{1}{3}\bigg)^2\cdot \cfrac{1}{3}+\bigg(\cfrac{4}{3}\bigg)^2\cdot \cfrac{1}{3}=\\
=\cfrac{42}{27}=\cfrac{14}{9}=1.556. σ 2 = ( 3 − 5 ) 2 ⋅ 3 1 + ( 3 1 ) 2 ⋅ 3 1 + ( 3 4 ) 2 ⋅ 3 1 = = 27 42 = 9 14 = 1.556.
The population standard deviation:
σ = 1.556 = 1.247. \sigma=\sqrt{1.556}=1.247. σ = 1.556 = 1.247.
The standard deviation of the sampling distribution of sample means:
σ x ˉ = σ n = 1.247 2 = 0.882. \sigma_{\bar x}=\cfrac{\sigma}{\sqrt n}=\cfrac{1.247}{\sqrt 2}=0.882. σ x ˉ = n σ = 2 1.247 = 0.882.
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