Final answer:
The mean number of correct answers for 30 multiple-choice questions with a 1 in 5 chance of guessing correctly is 6, and the standard deviation of the number of correct answers is 2.19. The correct selection from the given options is B) mean: 6; standard deviation: 2.19.
Step-by-step explanation:
To find the mean and the standard deviation of the number of correct answers for a multiple-choice test with 30 questions, we would consider each question to be a trial in a Bernoulli process since there are only two outcomes: a correct answer or an incorrect one. Since each question has five possible answers, the probability of getting a correct answer by randomly guessing is ⅓ or 0.2.
For a single question, the mean is the probability of getting it right, which is 0.2. Since there are 30 questions, we multiply that by 30 to get the mean number of correct answers, which is 30 × 0.2 = 6. This is the expected number of correct answers if one is guessing all answers.
The variance for a single question, which is a Bernoulli trial, is given by p(1 - p), where p is the probability of a success (getting the question right). So, the variance for one question is 0.2 × (1 - 0.2) = 0.16. The standard deviation is the square root of variance, so for one question it would be √0.16 = 0.4. However, since we have 30 independent questions, we multiply the variance by 30 before taking the square root to find the overall standard deviation. So, the standard deviation for 30 questions is √(30 × 0.16) = 2.19.
Therefore, the correct answer is B) mean: 6; standard deviation: 2.19.