Let X binom(n, ), (0, 1). Determinethefollowingapprox- imate confidenceintervalsfortheunknownprobabilityofsuccess : a) Aconfidenceintervalbasedonthequadraticscorestatistic Q(X,); b)

Let X ∼ binom(n, μ), μ ∈ (0, 1). Determinethefollowingapprox-

imate confidenceintervalsfortheunknownprobabilityofsuccess μ:

a) Aconfidenceintervalbasedonthequadraticscorestatistic Q(X,μ);

b) AconfidenceintervalbasedonaWaldstatisticforthelog-oddsratio

transformed backto μ.

c) AconfidenceintervalbasedonaWaldstatisticfortheparameter γ = sin−1(√
μ), transformed backto μ.

d) Makeasimulationstudytocomparetheseintervalswithrespecttocoverageand length.
Note thatthelength n in thebinomialdistributionof X can bothbeconsideredjust as alabel,butalsorepresentingthat X = Y1 + · ·· + Yn where Y1, ...,Yn are independent andidenticallyBernoullidistributed.Inthisway,theuseofasymptotic results isjustified,simplybecausetheFisherinformationtendstoinfinity.Compare this tothediscussioninExample5.41.

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0=log 1-