Question

At a certain college, \(6 \%\) of all students come from outside the United States. Incoming students there are assigned at random to freshman dorms, where students live in residential clusters of 40 freshmen sharing a common lounge area. How many international students would you expect to find in a typical cluster? With what standard deviation?

Short answer

Expected international students: 2.4; Standard deviation: 1.5.

Step-by-step solution

Determine the Probability

We start by identifying the probability of a student being international. Since 6% of all students are from outside the United States, this probability can be written as a decimal: \( p = 0.06 \).

Calculate the Expected Number

The expected number of international students in a cluster can be calculated using the formula for the expectation of a binomial distribution, \( E(X) = n \times p \), where \( n \) is the number of trials (students in the cluster). Here, \( n = 40 \). Thus, \( E(X) = 40 \times 0.06 = 2.4 \).

Calculate the Standard Deviation

The standard deviation of a binomial distribution is calculated using \( \sigma = \sqrt{n \times p \times (1-p)} \). Substituting the values, we get: \( \sigma = \sqrt{40 \times 0.06 \times 0.94} = \sqrt{2.256} \approx 1.5 \).

Key concepts

Probability

Understanding probability is essential for solving problems related to binomial distribution. In this problem, the probability that a student is international is given as 6%. We denote this probability as \( p \), which is a crucial component for further calculations. Remember that probabilities are always between 0 and 1. A probability of 0 means the event will never happen, while 1 means it certainly will happen. Here, to convert the percentage into a decimal format (which is the standard practice in probability calculations), we divide by 100. So, \( p = 0.06 \). In the context of binomial distribution, \( p \) represents the probability of success in a single trial. Each student being international is considered a "success", and since we're looking at one student at a time, each trial is independent of others. This understanding allows us to use binomial formulas for our problem.

Expected Value

The concept of expected value is like predicting the average outcome if the process were to be repeated many times. Using the binomial distribution formula, the expected value \( E(X) \) is given by \( E(X) = n \times p \). Here, \( n \) is the total number of students in the cluster, which is 40, and \( p \) is the probability of a student being international, 0.06. By plugging in these numbers: \[ E(X) = 40 \times 0.06 = 2.4 \] This means, in a typical dorm cluster of 40 students, you would expect to find about 2.4 international students, on average. The expected value doesn’t have to be a whole number because it is an average, not a count. This number gives us a central tendency of the distribution.

Standard Deviation

Standard deviation helps us understand the variability or spread of a data set. It tells us how much the actual data might deviate from the expected value.For a binomial distribution, the standard deviation \( \sigma \) is calculated by the formula \( \sigma = \sqrt{n \times p \times (1-p)} \). In our exercise, substituting the known values: \[ \sigma = \sqrt{40 \times 0.06 \times (1 - 0.06)} = \sqrt{2.256} \approx 1.5 \]This computation results in a standard deviation of approximately 1.5. This number gives us an approximation of how much the number of international students may vary from the expected 2.4. Emphasizing that while the expected count is around 2.4 international students, the actual number may fluctuate by around 1 to 2 students either side of this average, due to the natural variance in random samples.