Question

The probability of success on any trial of a binomial experiment is 25%. Find the probability that the proportion of successes in a sample of 500 is less than 22%.

Answer 1

Find the probability of success that the proportion of success in a sample of 500 is less than 22% .

The probability of success for any trail of a binomial experiment is, p=0.25.

The sample size, n = 500

The probability of success that the proportion of success in a sample of 500 is less than 22% is,

$P(\bar{P}<0.22)=P( \frac{\bar{P}-p}{\sqrt{\frac{p(1-p)}{n}}}<\frac{0.22-0.25}{\sqrt{\frac{0.25(1-0.25)}{500}}} ) \\ =P(Z<\frac{0.22-0.25}{0.0194}) \\ =p(Z<-1.55) \\ = 0.0606$

Therefore, the probability of success that the proportion of success in a sample of 500 is less than 22% is 0.0606.