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 500 is between 22% and 28%.
To find the probability that the proportion of successes in a sample of 500 is between 22% and 28%, we need to use the normal approximation to the binomial distribution.
The mean of the sample proportion is equal to the probability of success, which is 25%. The standard deviation of the sample proportion is calculated as:
standard deviation = sqrt((p * (1-p)) / n)
standard deviation = sqrt((0.25 * 0.75) / 500)
standard deviation = sqrt(0.1875 / 500)
standard deviation = sqrt(0.000375)
standard deviation = 0.01937
To find the probability that the proportion of successes is between 22% and 28%, we need to find the z-scores for these proportions:
z1 = (0.22 - 0.25) / 0.01937 = -1.55
z2 = (0.28 - 0.25) / 0.01937 = 1.55
Now, we look up the z-scores in the standard normal distribution table or use a calculator to find the probabilities associated with these z-scores:
P(0.22 < p < 0.28) = P(-1.55 < z < 1.55) = P(z < 1.55) - P(z < -1.55)
P(-1.55 < z < 1.55) = 0.9394 - 0.0606
P(-1.55 < z < 1.55) = 0.8788
Therefore, the probability that the proportion of successes in a sample of 500 is between 22% and 28% is 0.8788, or 87.88%.