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.

Question

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

Answer 1

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.

Answer 2

The number of trials is [?]

Answer 3

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.