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 less than 22%.
To solve this problem, we can use the Central Limit Theorem, which states that the sample proportion of successes in a binomial experiment converges to a normal distribution as the sample size increases.
First, we calculate the mean and standard deviation of the sample proportion of successes:
Mean = 0.25
Standard deviation = sqrt(p*(1-p)/n) = sqrt(0.25*0.75/1000) = 0.01369
Next, we standardize the sample proportion of successes:
Z = (0.22 - 0.25) / 0.01369 = -2.1914
Finally, we find the probability that the sample proportion of successes is less than 22% using a standard normal distribution table:
P(Z < -2.1914) ≈ 0.0143
Therefore, the probability that the proportion of successes in a sample of 1000 is less than 22% is approximately 1.43%.