Theoryexiststhatjustifiessubstitutingtheestimatedstandarderrorforthetrueoneinforming a pivotalquantity.Hereweshowthisforaproportion.

(a) The continuousmappingtheorem states thatcontinuousfunctionspreservelimits eveniftheirargumentsaresequencesofrandomvariables.Inparticular,if Xn p→

c and if g() is acontinuousfunctionat

c, then g(Xn) p→

g(c). Usingthis,explainwhy »

π(1 − π)~ˆπ(1 − ˆπ) p→

1.

(b) Slutsky’s Theorem states that if Zn d→

Z and if Xn p→

c, then XnZn d→

cZ. Apply this with Zn = (ˆπ − π)~

»

π(1 − π)~n and Xn =

»

π(1 − π)~ˆπ(1 − ˆπ) to justifythat (ˆπ −

π)~

»

ˆπ(1 − ˆπ)~n has alarge-samplestandard normal distribution.