Final answer:
To find the probability that the sample proportion will differ from the population proportion by less than 5%, we can use the normal distribution. The probability is approximately 0.9992.
Step-by-step explanation:
To find the probability that the sample proportion will differ from the population proportion by less than 5%, we can use the normal distribution. The sample size is 272 and the population proportion is 12% or 0.12. First, we need to calculate the standard deviation of the sample proportion, which can be found using the formula: √((p * (1-p)) / n), where p is the population proportion and n is the sample size. So, the standard deviation is √((0.12 * (1-0.12)) / 272), which is approximately 0.015.
Next, we need to calculate the difference between the sample proportion and the population proportion, and divide it by the standard deviation to find the z-score. In this case, the difference is 0.05, so the z-score is (0.05/0.015), which is approximately 3.33. We can then use a standard normal distribution table or a calculator to find the probability associated with this z-score, which is approximately 0.9992.
Therefore, the probability that the sample proportion will differ from the population proportion by less than 5% is 0.9992, rounded to four decimal places.