Question

The probability of success on any trail of a binomial experiment is 25%. Find the probability that the proportion of successes in a sample of 1000 is between 22% and 28%.

Answer 1

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

The sample size, n=1000

The probability of success that the proportion of success in a sample of 1000 is between 22% and 28% is,

$P(0.22<\bar{P}<0.28)=P( \frac{0.22-0.25}{\sqrt{\frac{0.25(1-0.25)}{1000}}} <\frac{\bar{P}-p}{\sqrt{\frac{p(1-p)}{n}}}<\frac{0.28-0.25}{\sqrt{\frac{0.25(1-0.25)}{1000}}} ) \\ =P(\frac{0.22-0.25}{0.0137}<Z<\frac{0.28-0.25}{0.0137}) \\ =P(-2.19<Z<2.19) \\ =\Phi(2.19)-\Phi(-2.19)\\ = 0.9857-0.0143=0.9714$

We found P using z-score table.