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

If both of np and nq are greater than or equal to 10, then the sampling distribution of sample proportions will be approximately normal with mean p and variance $\frac{pq}{n}$ .

$\hat{p}~ N(P, \frac{pq}{n})$

p = population proportion,

q = 1 - p

n = sample size

p = 0.55

q = (1 - 0.55) = 0.45

n = 500

np = 275, which is greater than 10.

nq = 225, which is greater than 10.

Hence sampling distribution of sample proportions will be approximately normal.

We have to find P($\hat{p} < 0.49$ ).

$Z = \frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} \\ P(\hat{p}<0.49) = P( \frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}} < \frac{0.49-p}{\sqrt{ \frac{pq}{n} }}) \\ = P(Z< \frac{0.49-0.55}{\sqrt{ \frac{0.55 \times 0.45}{500} }}) \\ = P(Z< -2.696) \\ = 0.035$