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 800 is between 22% and 28%.
To find the probability that the proportion of successes in a sample of 800 is between 22% and 28%, we will use the normal approximation to the binomial distribution.
First, we need to calculate the mean and standard deviation of the binomial distribution:
Mean (μ) = n * p = 800 * 0.25 = 200
Standard deviation (σ) = sqrt(n * p * (1 - p)) = sqrt(800 * 0.25 * 0.75) = 10
Next, we need to standardize the range of proportions:
Z1 = (0.22 - 0.25) / 0.1 = -0.30
Z2 = (0.28 - 0.25) / 0.1 = 0.30
Now, we will use a z-table to find the probability:
P(0.22 < p < 0.28) = P(-0.30 < Z < 0.30)
P(-0.30 < Z < 0.30) = P(Z < 0.30) - P(Z < -0.30)
P(Z < 0.30) = 0.6179
P(Z < -0.30) = 0.3821
Therefore, P(-0.30 < Z < 0.30) = P(Z < 0.30) - P(Z < -0.30) = 0.6179 - 0.3821 = 0.2358
So, the probability that the proportion of successes in a sample of 800 is between 22% and 28% is 23.58%.