We have population values 4, 5, 8, 12, 9, 12, population size N=6 and sample size n=4.
a. Mean of population (μ) = 64+5+8+12+9+12=25/3
b. Variance of population
σ2=NΣ(xi−xˉ)2=986
σ=σ2=986=386
c. The number of possible samples which can be drawn without replacement is NCn=6C4=15.
no123456789101112131415Sample4,5,8,124,5,8,94,5,8,124,5,12,94,5,12,124,5,9,124,8,12,94,8,12,124,8,9,124,12,9,125,8,12,95,8,12,125,8,9,125,12,9,128,12,9,12Samplemean (xˉ)29/426/429/430/433/430/433/436/433/437/434/437/434/438/441/4
Xˉ26/429/430/433/434/436/437/438/441/4f(Xˉ)1/152/152/153/152/151/152/151/151/15Xˉf(Xˉ)26/6058/6060/6099/6068/6036/6074/6038/6041/60Xˉ2f(Xˉ)676/2401682/2401800/2403267/2402312/2401296/2402738/2401444/2401681/240
Mean of sampling distribution
μXˉ=E(Xˉ)=∑Xˉif(Xˉi)=25/3
d. The variance of sampling distribution
Var(Xˉ)=σXˉ2=∑Xˉi2f(Xˉi)−[∑Xˉif(Xˉi)]2=24016896−(325)2=4543
e.
σXˉ=4543≈0.9775
f.
μXˉ=E(Xˉ)=25/3=μ
Var(Xˉ)=σXˉ2=4543=nσ2(N−1N−n)
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