1)A quiz consists of 8 multiple choice questions, each with 5 possible answers.
a. If a student guesses on all 8 of the questions, do the number of questions
answered correctly form a binomial distribution?
A. Yes B. No
b. If "yes", give the mean (µ) and standard deviation (σ), if "no" give at least 1
requirement of a binomial distribution that is not met.
a. Yes, the number of questions answered correctly forms a binomial distribution. In order for a situation to be considered a binomial distribution, it must meet the following criteria:
1) There are a fixed number of trials: In this case, the fixed number of trials is 8 questions.
2) Each trial has two possible outcomes, usually labeled as success (answered correctly) or failure (answered incorrectly).
3) The probability of success, denoted as p, is constant for each trial.
4) The trials are independent of each other, meaning the outcome of one trial does not affect the outcome of any other trial.
In this case, when a student guesses on all 8 questions, each question has a 1/5 probability of being answered correctly. Since all criteria for a binomial distribution are met, we can conclude that the number of questions answered correctly forms a binomial distribution.
Answer: A. Yes
b. Since the number of questions answered correctly forms a binomial distribution, we can determine the mean (µ) and standard deviation (σ) using the following formulas:
Mean (µ) = n * p
Standard Deviation (σ) = √(n * p * (1 - p))
where n is the number of trials (8) and p is the probability of success (1/5).
Mean (µ) = 8 * (1/5) = 1.6
Standard Deviation (σ) = √(8 * (1/5) * (1 - 1/5)) = √(8 * (1/5) * (4/5)) = √(8 * 4/25) = √(32/25) ≈ 0.9
So, the mean is 1.6 and the standard deviation is approximately 0.9.
Answer: Yes, the number of questions answered correctly forms a binomial distribution with a mean of 1.6 and a standard deviation of approximately 0.9.