Question

Clarinex-D is a medication intended to reduce symptoms associated with allergies. In clinical trials, 5% of patients experienced insomnia as a side effect.

If 240 users of Clarinex-D are randomly selected:

(a) Let X = number of people in the sample who experience insomnia as a side effect. Give the values of n and p for this binomial random variable.

(b) What is the expected number of people in the sample who experience insomnia as a side effect?

(c) The value of p is fairly small (not close to 0.5). Show that our sample is large enough for the distribution of the random variable X to be approximately bell-shaped. (Remember the rule of thumb!)

(d) Based on the empirical rule, what range of values should contain 68% of the values of X?

(e) Would a sample with 30 people suffering from the insomnia side effect be considered unusual?

Answer 1

(a). Here n=240 and p=0.05.

(b). We know that in Binomial distribution, the mean is given by n*p=240*0.05=12

(c).The rule of Thumb says that n*p>5 and n(1-p).5. Here np=12 and n(1-p)=240*0.95=228. Hence the sample is large enough.

(d). We know that by Empirical rule, 68% of the data falls within 1 sd. Hence, the Variance of a Binomial distribution is

n*p*(1-p)=240*0.05*0.95=11.4. The standard deviation is sd=

Therefore

=(8.6236,15.3764)

The range of values is therefore =(9,15) after rounding off the numbers.

(e). Yes, a sample of 30 people with insomnia is unusual since p=0.125 =12.5% which is quite high compared with the 5%.