Answer to Question #219306 in Statistics and Probability for Loo

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($\hat{p} < 0.49$ ).

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