Compute the Mean and Standard Deviation of a Binomial Random Variable
-A quiz consists of 20 multiple choice questions, each with five possible answers, only one of which is correct. Find the mean and the standard deviation of the number of correct answers.
A) mean: 4; standard deviation: 1.78885438
B) mean: 4; standard deviation: 2
C) mean: 10; standard deviation: 1.78885438
D) mean: 10; standard deviation: 3.16227766

Question

Quizplus · Accepted Answer

The mean of a binomial distribution is given by $ \mu = np $, where $ n $ is the number of trials and $ p $ is the probability of success on each trial. Here, $ n = 20 $ and $ p = \frac{1}{5} $ because there is one correct answer out of five options for each question. Thus, the mean is $ 20 \times \frac{1}{5} = 4 $. The standard deviation is given by $ \sigma = \sqrt{np(1-p)} $, which calculates to $ \sqrt{20 \times \frac{1}{5} \times \frac{4}{5}} = \sqrt{16} = 4 $. However, there was a mistake in my calculation; the correct standard deviation should be $ \sqrt{20 \times \frac{1}{5} \times \frac{4}{5}} = \sqrt{3.2} \approx 1.78885438 $.