Question: Consider n independentobservationsofarandomvariable Y that hasskewnesscoefficient S = E(Y )3~3. (a) Showhow E(Y )3 relates to E(Y )3. Based on this,showthattheskewnesscoefficient for thesamplingdistributionof Y

Consider n independentobservationsofarandomvariable Y that hasskewnesscoefficient S = E(Y − μ)3~σ3.

(a) Showhow E(Y −μ)3 relates to E(Y −μ)3. Based on this,showthattheskewnesscoefficient for thesamplingdistributionof Y satisfies skewness(Y ) = S~

n. Explainhowthis result relates totheCentralLimitTheorem.

(b) Supposethatwecanselectavalue M suchthatweregardaunimodaldistributionhaving S < M as being“close”tobell-shaped.Withsimplerandomsampling,showthatthe sampling distributionof Y is thenapproximatelynormal if n > (S~M)2.

(c) PlotsofcommonunimodaldistributionssuchasthePoissonandgammawithvarious valuesof S suggest that S ≤ M ≈

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