Final answer:
The exact probability of scoring a 5 on the test by guessing is very low, approximately 0.000034. The approximate probability using the normal distribution is approximately 99.65%.
Step-by-step explanation:
The probability of guessing the correct answer to a multiple choice question with 5 total answers is 1/5 or 0.2. Since there are 11 questions on the test, the probability of guessing all 11 correctly is (1/5)^11 or approximately 0.000034. This is the exact probability of scoring a 5 on the test by guessing.
To approximate the probability using the normal distribution, we can use the formula for the mean and standard deviation of a binomial distribution.
The mean is given by np, where n is the number of trials (11) and p is the probability of success (0.2). The standard deviation is given by sqrt(npq), where q is the probability of failure (1-p).
Using these formulas, we find that the mean is 11 * 0.2 = 2.2 and the standard deviation is sqrt(11 * 0.2 * 0.8) ≈ 1.49. To find the probability of scoring a 5 or higher, we can use the normal distribution with continuity correction.
We calculate the z-score for 4.5 (halfway between 4 and 5) using the formula z = (x - mean) / standard deviation. With a z-score of 2.68, we can look up the corresponding probability in the standard normal distribution table, which is approximately 0.9965 or 99.65%.