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 more than 28%.
To solve this problem, we can use the normal approximation to the binomial distribution since the sample size is large (n = 1000) and the probability of success is small (p = 0.25).
First, we need to find the mean and standard deviation of the sample distribution. The mean of the sample distribution is given by:
mean = n*p = 1000 * 0.25 = 250
The standard deviation of the sample distribution is given by:
standard deviation = sqrt(n * p * (1 - p)) = sqrt(1000 * 0.25 * 0.75) = sqrt(187.5) ≈ 13.70
Next, we need to standardize the value of 28% using the z-score formula:
z = (x - mean) / standard deviation
z = (28 - 250) / 13.70
z ≈ -17.42
To find the probability that the proportion of successes in a sample of 1000 is more than 28%, we need to find the area to the right of z = -17.42 on the standard normal distribution. This area represents the probability of getting more than 28% successes in the sample.
Using a standard normal table or a calculator, we can find that the probability of getting more than 28% successes in the sample is essentially 0 (rounded to 0.0000).
Therefore, the probability that the proportion of successes in a sample of 1000 is more than 28% is approximately 0.