Question

Clarinex-D is a medication whose purpose is to reduce the symptom associated with a variety of allergies. In a clinical trial of Clarinex-D, 5% of the patients in the study experience insomnia as a side effect. A random sample 20 Clarinex-D users is selected and the number of patients who experience insomnia is recorded.

A) Find the probability that exactly 3 experience insomnia as a side effect?
B) Find the probability 3 or fewer experience insomnia?
C) Between 1 and 4 experience insomnia.
D) Would it be unusual to find 4 or more patients experience insomnia as a side effect?

Answer 1

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.