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%.
0.8866
0.9714
0.8413
0.9287
0.9972
0.8866
This can be calculated using the central limit theorem and the normal distribution approximation for the binomial distribution. The mean of the sample proportion is equal to the true probability of success (0.25), and the standard deviation is calculated as sqrt(p(1-p)/n) = sqrt(0.25*0.75/1000) = 0.01369.
To find the probability that the sample proportion is between 22% and 28%, we need to find the z-scores for these values:
z1 = (0.22 - 0.25) / 0.01369 = -2.1922
z2 = (0.28 - 0.25) / 0.01369 = 2.1919
Using a standard normal table or calculator, we can find the probabilities corresponding to these z-scores:
P(z < -2.1922) = 0.0141
P(z < 2.1919) = 0.9859
Therefore, the probability that the sample proportion is between 22% and 28% is P(0.22 < p < 0.28) ≈ 0.9859 - 0.0141 = 0.9718.