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%
To find the probability that the proportion of successes in a sample of 1000 is between 22% and 28%, we need to use the normal approximation to the binomial distribution.
First, we need to calculate the mean and standard deviation of the binomial distribution. The mean of a binomial distribution is np, where n is the sample size and p is the probability of success. In this case, n = 1000 and p = 0.25, so the mean is 1000 * 0.25 = 250.
The standard deviation of a binomial distribution is sqrt(np(1-p)). In this case, the standard deviation is sqrt(1000 * 0.25 * 0.75) = 13.69.
Next, we need to standardize the values of 22% and 28% to z-scores using the formula z = (x - mean)/standard deviation. For x = 22%:
z = (0.22 * 1000 - 250) / 13.69 ≈ -1.825
For x = 28%:
z = (0.28 * 1000 - 250) / 13.69 ≈ -0.729
Now we can use a standard normal table or a calculator to find the area under the standard normal curve between z = -1.825 and z = -0.729. This represents the probability that the proportion of successes in a sample of 1000 is between 22% and 28%.
Using a standard normal table or a calculator, the probability comes out to be approximately 0.153. So, the probability that the proportion of successes in a sample of 1000 is between 22% and 28% is 15.3%.