Question

Experience has shown that 30% of all persons afflicted by a certain illness will recover with the standard treatment that is available. A drug company has developed a new therapy for this illness. Fifteen people with the illness are selected at random and treated with the new therapy. Eight of these fifteen recover from the illness.

a. Suppose that the new therapy is as effective as the standard treatment. Under this assumption, what is the probability that 8 or more patients out of 15 treated by the new therapy would recover?

b. What assumptions are you making about your data and the study design to compute this probability?

c. Under this model, what is the expected number of patients who will recover? The standard deviation?

d. Do you think that the proportion cured with the new drug is better than the old? Why or why not? Use a statistical argument to support your conclusion.

Answer 1

As per binomial distribution,
P(X=r) = nCr * p^r * (1-p)^(n-r)
Here, n = 15, p = 0.30

a)

P(x > =8) = 1 - P(x < =7)
= 1 - 0.95
= 0.05

b)

1) Each replication of the process results in one of two possible outcomes (success or failure),
2) The probability of success is the same for each replication, and
3) The replications are independent, meaning here that a success in one patient does not influence the probability of success in another.

c)

mean = np = 15 * 0.30 = 4.5

std.dev = sqrt(npq)
= sqrt(15 * 0.30 * 0.70)

=1.7748