Answer to Question #219321 in Statistics and Probability for Maseko

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 i 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 $p_1=0.49$

$$0.49(500)=245$$

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

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