A study at a college on the west coast reveals that, historically, 45% of the students are minority students. If a random sample of size 75 is selected, the probability is ________ that more than half of the students in the sample will be minority students.
To find the probability that more than half of the students in the sample will be minority students, we first need to determine the probability of exactly half of the students being minority students, and then add the probabilities of more than half of the students being minority students.
The probability of exactly half of the students in the sample being minority students can be calculated using the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where:
n = sample size = 75
k = number of minority students = 37.5 (half of 75)
p = probability of being a minority student = 0.45
Plugging in the values, we get:
P(X = 37.5) = (75 choose 37.5) * (0.45)^37.5 * (0.55)^37.5
Now, we need to calculate the probability of more than half of the students being minority students:
P(X > 37.5) = 1 - P(X <= 37.5)
P(X <= 37.5) = sum of P(X = 37.5) + P(X = 0) + P(X = 1) + ... + P(X = 37)
This calculation needs to be performed using a statistical software or a binomial probability table to obtain the final probability.