Final answer:
To find the probabilities in this clinical trial, we can use the binomial probability formula. Using the formula, we can find the probability of exactly 3, 3 or fewer, and 1 to 4 people experiencing insomnia. By calculating these probabilities, we determine that it is unusual for 4 or more people to experience insomnia as a side effect of this medication.
Step-by-step explanation:
To answer these questions, we can use the binomial probability formula. The formula is:
P(X = k) = C(n,k) * p^k * (1-p)^(n-k)
where P(X = k) is the probability of exactly k successes, C(n,k) is the number of combinations of n items taken k at a time, p is the probability of success on a single trial, and n is the number of trials.
A) For exactly 3 people to experience insomnia, we substitute k = 3 into the formula:
P(X = 3) = C(20,3) * 0.05^3 * (1-0.05)^(20-3)
Plugging the values into a calculator, we find that the probability is approximately 0.2706.
B) For 3 or fewer people to experience insomnia, we need to find the cumulative probability:
P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Calculating each individual probability and summing them up, we find that the probability is approximately 0.9940.
C) To find the probability that between 1 and 4 people experience insomnia, we need to sum the probabilities for X = 1, X = 2, X = 3, and X = 4:
P(1 ≤ X ≤ 4) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
Calculating each individual probability and summing them up, we find that the probability is approximately 0.9789.
D) To determine if it is unusual for 4 or more people to experience insomnia, we can calculate the probability of that happening:
P(X ≥ 4) = 1 - P(X ≤ 3)
Calculating the probability of X ≤ 3 as calculated in part B, we find that the probability of 4 or more people experiencing insomnia is approximately 0.0060.