Suppose that 40% of all college students smoke cigarettes. A sample of 20 is selected randomly. What is the probability that at most a fourth of the students smoke?
To find the probability that at most a fourth of the students smoke, we need to calculate the cumulative probability from zero up to a fourth.
Given that 40% of all college students smoke cigarettes, we can assume that the smoking rate follows a binomial distribution. The formula to calculate the probability of getting exactly k successes (students who smoke) in n trials (total sample) with a probability of success p (proportion of students who smoke) is given by the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k),
where C(n, k) is the binomial coefficient, also known as "n choose k" and can be calculated as C(n, k) = n! / (k! * (n-k)!).
In this case, we want to find the probability P(X <= 1/4), which means either 0 or 1 student in our sample smokes. So we need to calculate P(X = 0) and P(X = 1), and then sum them up.
First, let's calculate P(X = 0):
n = 20 (total sample size)
k = 0 (no students smoke)
p = 0.40 (proportion of students who smoke)
P(X = 0) = C(20, 0) * 0.40^0 * (1 - 0.40)^(20 - 0)
P(X = 0) = 1 * 1 * 0.60^20
P(X = 0) = 0.60^20 ≈ 0.026
Next, let's calculate P(X = 1):
n = 20 (total sample size)
k = 1 (one student smokes)
p = 0.40 (proportion of students who smoke)
P(X = 1) = C(20, 1) * 0.40^1 * (1 - 0.40)^(20 - 1)
P(X = 1) = 20 * 0.40^1 * 0.60^19 ≈ 0.106
Now, we can sum up these probabilities to find the cumulative probability:
P(X <= 1/4) = P(X = 0) + P(X = 1)
P(X <= 1/4) ≈ 0.026 + 0.106 ≈ 0.132
Therefore, the probability that at most a fourth of the students smoke is approximately 0.132 or 13.2%.