Question

The proportion of eligible voters in the next 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 $p_1=0.49$

$$0.49(500)=245$$

$$P(X<0.49)=P(Z<\dfrac{245-275}{\sqrt{123.75}})$$

$$\approx P(Z<-2.6928)\approx0.0035$$