Question

The proportion of eligible voters in the next corporation election (in tumkur) who will vote for the incumbent is assumed to be 55%. What is the probability that in a random sample of 500 voters less than 49% say they will vote for the incumbent? Write your conclusion.

Answer 1

$P\left( \hat{p}\leqslant 0.49 \right) =P\left( \sqrt{n}\frac{\hat{p}-p}{\sqrt{p\left( 1-p \right)}}\leqslant \sqrt{n}\frac{0.49-p}{\sqrt{p\left( 1-p \right)}} \right) =\\=P\left( Z\leqslant \sqrt{500}\frac{0.49-0.55}{\sqrt{0.55\cdot 0.45}} \right) =\varPhi \left( -2.6968 \right) =0.00350$

The event is of low probability.