Question

Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, 5% of the patients in the study experienced insomnia as a side effect. As random sample of 20 Clarinex-D users is obtained, and the number of patients who experienced insomnia is recorded. Find and interpret probability that 3 or fewer experienced insomnia as a side effect.

Answer 1

Using the binomial distribution, it is found that there is a 0.9842 = 98.42% probability that 3 or fewer experienced insomnia as a side effect, which means that it is a highly likely event.

What is the binomial distribution formula?

The formula is:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• n is the number of trials.

• p is the probability of a success on a single trial.

The values of the parameters are given as follows:

n = 20, p = 0.05.

The probability that 3 or fewer experienced insomnia as a side effect is given by:

$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

Hence:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{20,0}.(0.05)^{0}.(0.95)^{20} = 0.3585$

$P(X = 1) = C_{20,1}.(0.05)^{1}.(0.95)^{19} = 0.3774$

$P(X = 2) = C_{20,2}.(0.05)^{2}.(0.95)^{18} = 0.1887$

$P(X = 3) = C_{20,3}.(0.05)^{3}.(0.95)^{17} = 0.0596$

Then:

$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.3585 + 0.3774 + 0.1887 + 0.0596 = 0.9842$

0.9842 = 98.42% probability that 3 or fewer experienced insomnia as a side effect.

Since this probability is greater than 95%, this is a highly likely event.

More can be learned about the binomial distribution at https://brainly.com/question/24863377

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