Final answer:
To calculate the number of students expected to fail on the CPA examination, multiply the total number of students by the failure rate. The standard deviation can be calculated using a formula, and the probabilities of exactly six students failing and at least six students failing can be calculated using the binomial distribution.
Step-by-step explanation:
To calculate the number of students expected to fail, we need to multiply the total number of students taking the examination by the percentage of students who fail. In this case, there are 79 students taking the exam and the failure rate is 15%. Therefore, we expect approximately 11.85 students to fail (79 * 0.15 = 11.85).
The standard deviation can be calculated using the formula:
sqrt(n * p * (1-p)), where n is the sample size and p is the probability of success. In this case, the sample size is 79 and the probability of success is 1 - 0.15 = 0.85. Therefore, the standard deviation is approximately 4.16.
To calculate the probability that exactly six students will fail, we can use the binomial distribution formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where n is the sample size, k is the number of successes, and p is the probability of success. In this case, n = 79, k = 6, and p = 0.15. Using the formula, we can calculate the probability to be approximately 0.183.
To calculate the probability that at least six students will fail, we need to calculate the probability of six, seven, eight, ..., 79 students failing and then add all those probabilities together. Alternatively, we can calculate the complementary probability (probability that fewer than six students fail) and subtract it from 1. Using the binomial distribution formula, we can calculate the probability of fewer than six students failing to be approximately 0.952. Therefore, the probability of at least six students failing is approximately 1 - 0.952 = 0.048.