Question

A drug manufacturer markets a brand of pain reliever that it claims to be 95% effective in relieving headache pain. Assume that the effect of the drug on each person is independent of the effect on all other people receiving the drug, and the 95% effective rate is the same for each person with headache pain.
A consumer group decides to select a sample of 20 people at random who suffer from headache pain and give them the medicine. If the number of people in the sample who experience pain relief is 18 or less, the consumer group will publish a report questioning the claim that the drug is 95% effective; otherwise, no report of the results will be published. Please answer the following questions. (Hint: think about the Binomial distribution)
(a) If p=.95 as the manufacturer claims, what is the expected number of people in the sample who will experience pain relief?
(b) If p=.95 as the manufacturer claims, what is the standard deviation the number of people in the sample who will experience pain relief?
(c) If p=.95 as claimed, what is the probability that the sample of 20 people will have 18 or fewer who experience pain relief?

Answer 1

Solution :

Let X be a random variable which represents the number of people in a group of 20 people who will experience pain releif on receiving the pain reliever.

It is claimed that, pain reliever is 95% effective. Hence, probability that an person will experience the relief in pain on receiving the pain reliever is, 95/100 = 0.95.

If we consider "a person who experience pain releif" as success, then we have only two mutually exclusive outcomes (success and failure).

Probability of success (p) = 0.95

Number of trials (n) = 20

Since, probability of success remains constant in each of the trials, number of trials are finite, we have only two mutually exclusive outcomes for each of the trials and outcomes are independent to each other, therefore we can consider that X follows binomial distribution with parameters n = 20 and p = 0.95

a) The number of people in the sample who will experience pain relief would follows binomial distribution.

The expected value for binomial distribution is given as follows :

E(X) = np

We have, n = 20 and p = 0.95

Hence, E(X) = 20 × 0.95 = 19

Hence, 19 people are expected to experience pain relief in the sample.

b) The number of people in the sample who will experience pain relief would follows binomial distribution.

The standard deviation for binomial distribution is given as follows :

We have, n = 20 and p = 0.95

Hence, the standard deviation is 0.9747.

c) We have to obtain P(X ≤ 18).

We have, n = 20, p = 0.95

According to binomial probability law, the probability of occurrence of exactly x successes in n trials is given by,

P(X ≤ 18) = 1 - P(X > 18)

P(X ≤ 18) = 1 - [P(X = 19) + P(X = 20)]

Using binomial probability law we get,

Hence, the probability that the sample of 20 people will have 18 or fewer who experience pain relief is 0.2642.

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