Question
Determine whether the random variable X has a binomial distribution. If it does, state the number of trials N. If it does not, explain why not.
Thirty students are randomly chosen from a small college with an enrollment of 2400. Let X be the number who are freshmen students.
The random variable [blank] a binomial distribution
Thirty students are randomly chosen from a small college with an enrollment of 2400. Let X be the number who are freshmen students.
The random variable [blank] a binomial distribution
Answers
Answer
The number of trials is [?]
Answered by
GPT-4o mini
To determine whether the random variable \( X \) (the number of freshmen students among the 30 randomly chosen students) has a binomial distribution, we need to check if it meets the conditions for a binomial distribution:
1. **Fixed number of trials (N)**: We have a fixed number of trials, which is \( N = 30 \) (the number of students chosen).
2. **Two possible outcomes**: Each student can either be a freshman or not.
3. **Constant probability of success (p)**: For a binomial distribution, it is assumed that the probability \( p \) of success (selecting a freshman) is constant for all trials. However, since we are choosing students from a finite population without replacement (the college has only 2400 students), the probability changes slightly with each selection if we don't use the hypergeometric distribution.
4. **Independence**: For binomial distributions, the trials must be independent. In this case, because we are sampling without replacement from a finite population, the trials are not independent. The outcome of one trial influences the outcomes of others.
Since the probability of selecting a freshman changes as students are selected (due to the finite population and sampling without replacement), and because the trials are not independent, the random variable \( X \) does not meet the requirements for a binomial distribution.
Thus, we can conclude:
The random variable \( X \) does **not** have a binomial distribution because the trials are not independent and the probability of success is not constant throughout the selection process. This scenario could be modeled using a hypergeometric distribution instead.
1. **Fixed number of trials (N)**: We have a fixed number of trials, which is \( N = 30 \) (the number of students chosen).
2. **Two possible outcomes**: Each student can either be a freshman or not.
3. **Constant probability of success (p)**: For a binomial distribution, it is assumed that the probability \( p \) of success (selecting a freshman) is constant for all trials. However, since we are choosing students from a finite population without replacement (the college has only 2400 students), the probability changes slightly with each selection if we don't use the hypergeometric distribution.
4. **Independence**: For binomial distributions, the trials must be independent. In this case, because we are sampling without replacement from a finite population, the trials are not independent. The outcome of one trial influences the outcomes of others.
Since the probability of selecting a freshman changes as students are selected (due to the finite population and sampling without replacement), and because the trials are not independent, the random variable \( X \) does not meet the requirements for a binomial distribution.
Thus, we can conclude:
The random variable \( X \) does **not** have a binomial distribution because the trials are not independent and the probability of success is not constant throughout the selection process. This scenario could be modeled using a hypergeometric distribution instead.
Answered by
GPT-4o mini
The number of trials \( N \) is 30, as it represents the number of students being randomly chosen from the college.
So, to summarize:
The number of trials is \( N = 30 \).
However, please note that while \( X \) describes the number of freshmen students out of these trials, \( X \) does not have a binomial distribution due to the reasons provided previously. It should be modeled as a hypergeometric distribution instead.
So, to summarize:
The number of trials is \( N = 30 \).
However, please note that while \( X \) describes the number of freshmen students out of these trials, \( X \) does not have a binomial distribution due to the reasons provided previously. It should be modeled as a hypergeometric distribution instead.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.