Answer to Question #219297 in Statistics and Probability for Rato

Question

The proportion of eligible voters in the election who will vote for the ANC is assumed to be 0.55 in Gauteng. What is the probability that in a random of 500 Gauteng voters less than 0.49 say they will vote for the ANC?

Answer 1

Let $X$ be a binomial rv based on $n$ trials with success probability $p.$ Then if the binomial probability histogram is not too skewed, $X$ has approximately a normal distribution with $\mu=np$ and $\sigma^2=npq.$

In practice, the approximation is adequate provided that both $np\geq 10$ and $nq\geq 10.$

Given $n=500, p=0.55, q=1-p=1-0.55=0.45.$

Check $np=500(0.55)=275>10, nq=500(0.45)=225>10.$

Then $\mu=np=275, \sigma^2=500(0.55)(0.45)=123.75.$

$X\sim N(275, 123.75)$

Given $p_1=0.49$

$$0.49(500)=245$$ $$P(X<0.49)=P(Z<\dfrac{245-275}{\sqrt{123.75}})$$ $$\approx P(Z<-2.6928)\approx0.0035$$