To use the Central Limit Theorem for Sample Proportions, we need to verify the following conditions are met:
1. Random Sample: We are told that the sample is random.
2. Independence: The sample size is less than 10% of the population size, so we can assume independence.
3. Success/Failure Condition: The expected number of successes and failures are both at least 10. The expected number of successes is 0.41 * 300 = 123 and the expected number of failures is 177, so this condition is met.
Now, we can use the Central Limit Theorem to approximate the sample proportion distribution with a normal distribution with mean = p = 0.41 and standard deviation = sqrt((p*(1-p))/n) = sqrt((0.41*0.59)/300) = 0.032.
To find the probability that at most 39% of the sample have a BA degree, we need to find the probability that the sample proportion is less than or equal to 0.39.
Using the normal distribution with mean = 0.41 and standard deviation = 0.032, we can standardize the sample proportion as follows:
z = (0.39 - 0.41) / 0.032 = -0.625
Using a standard normal distribution table, we can find that the probability of getting a z-score less than or equal to -0.625 is approximately 0.266.
Therefore, the probability that at most 39% of the sample have a BA degree is approximately 0.266.