Question

It was claimed that 1 of 5 cardiologists take aspirin a day to prevent hardening of the arteries. Suppose that the claim is true. If 16 cardiologists are selected independently at random, let X be the number who take aspirin a day. (a) How is X distributed? (b) Give the values of E(x) and Var (x) (c) Determine P(X>5)

Answer 1

Solution:

a) X follow the binomial distribution with n=16 and probability of success p= 1/5= 0.2

b) E(X)=np= 16*0.2= 3.2 and

V(X)= np(1-p)= 16*0.2*0.8= 2.56

C) P(X > 5)= 1 - P(X ≤ 4) = 1 - 0.7982= 0.2018

Calculation:

The following information is provided: The population proportion of success is p = 0.2, also, 1-p=1 – 0.2 = 0.8, and the sample size is n = 16. We need to compute Pr(X < 4) Therefore, we get that Pr(X < 4) =) Pr(X < 4) = ÝP+(x = 1) = ('a –p) -i (X = i)= - p)n-i i=0 This implies that Pr(X < 4) = Pr(X = 0) + Pr(X = 1) + Pr(X = 2) + Pr(X = 3) + Pr(X = 4) = (15) 0.20 x 0.916-0 + (18) 0.21 x 0.816-1 + (19) 0.22 x 0.826–2 + (15) 0.28 x 0.816-3 + (16) 0.24 x 0.916-4 = 0.0281 + 0.1126 + 0.2111 +0.2463 + 0.2001 = 0.7982 which means that the probability we are looking for is Pr(X < 4) = 0.7982.