Question

A multiple choice exam consists of 10 questions. Questions 1-5 have 5 alternatives for each question and Questions 6-10 have 7 alternatives for each question. A student chooses an answer at random for each question. X is the number of correct choices made by the student for questions 1-5. Y is the number of correct choices made by the student for questions 6-10. Z is the number of correct choices made by the student in the exam. ( so Z X+Y) Then

a. X is binomial but Y and Z are not
b. X, Y, Z are all binomial
c. Y is binomial but X and Z are not
d. X, Y are binomial but Z is not
e. none of X, Y, Z is a binomial

Answer 1

X and Y are binomial variables as they meet the criteria for binomial distribution, with fixed numbers of independent trials and consistent probabilities of success. Z, being the sum of X and Y with different probabilities of success, does not follow a binomial distribution.

Step-by-step explanation:

To determine whether X, Y, and Z variables in a multiple-choice exam follow a binomial distribution, we need to consider the characteristics of the binomial distribution:

• There must be a fixed number of trials.
• There must be only two possible outcomes (success or failure) for each trial.
• The trials must be independent, and the probability of success (p) stays the same for each trial.

Variable X counts the correct answers for questions 1-5, where each question has 5 alternatives, making the probability of guessing correctly 1/5. There are 5 questions, so there are 5 trials. Variable Y counts the correct answers for questions 6-10, where each question has 7 alternatives, making the probability of guessing correctly 1/7. There are 5 questions, so there are also 5 trials. Therefore, both X and Y are binomial variables because they meet the criteria for a binomial distribution. However, Z is the sum of two binomial distributions with different probabilities of success (1/5 for X and 1/7 for Y), which means Z doesn't follow a binomial distribution because it doesn't have a single, consistent probability of success across all trials. Hence, the answer is that X and Y are binomial, but Z is not.