Answer to Question #315043 in Statistics and Probability for heet

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.