In a large Midwestern university, 30% of the students live in apartments. If 200 students are randomly selected, find the probability that the number of them living in apartments will be between 55 and 70 inclusive.
To find the probability that the number of students living in apartments will be between 55 and 70 inclusive, we need to use the binomial probability formula.
The binomial probability formula is given by:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
where:
P(X = k) is the probability of exactly k successes
n is the total number of trials (students selected)
k is the number of successes (students living in apartments)
p is the probability of success (students living in apartments)
nCk is the binomial coefficient, also known as "n choose k"
First, let's calculate the values we need:
n = 200 (total number of students selected)
k = 55, 56, 57, ..., 70 (the range of successes, number of students living in apartments)
p = 0.30 (probability of a student living in an apartment)
Now, we can plug these values into the binomial probability formula for each value of k and sum up the probabilities for all the values of k between 55 and 70 inclusive:
P(55 <= X <= 70) = P(X = 55) + P(X = 56) + P(X = 57) + ... + P(X = 70)
= [(200C55) * 0.30^55 * 0.70^145] + [(200C56) * 0.30^56 * 0.70^144] + [(200C57) * 0.30^57 * 0.70^143] + ... + [(200C70) * 0.30^70 * 0.70^130]
You can calculate the values using a calculator or statistical software that supports binomial calculations.