DATE 7. The probobility that a person is intected by a certain cold vinus is 0.2

Question

DATE 7. The probobility that a person is intected by a certain cold vinus is 0.25. A random sample of 12 people is taken. (6 pts) Use the binomial probability model to determine what the probability is that no more than 2 people will be infected by this cold virus. a. b. (3 pts) Find the sample mean C. (3 pts) Find the sample standard deviation d. l6 pts) Use the normal approximation to estimate the probobility in part a e. 16 pts) Find the probability that at most 11 people have the virus. f. (6 pts) Find the probability that 2 people have the virus given than at least one person has the virus. Page 6 of 15 Final Exam Math 104-03 (Spring 2018)

Explanation / Answer

p = 0.25 , n = 12

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

a)
P(x < 2) = P(x = 0) + P(x =1)
= 12C0 * 0.25^0 * (1-0.25)^12 + 12C1 * 0.25^1 * (1-0.25)^11
= 0.1584

b)
sample mean = n p = 12 * 0.25 = 3

c)
sample std.deviation = sqrt(npq)
= sqrt(12 * 0.25 * 0.75) = 1.5

d)

P(x <2)

z = (x -mean)/s
= (2-3)/1.5
= -0.6667

P(x <2) = P(z < -0.6667) = 0.2525

e)

P(x < =11) = P(x = 0) + P(x =1 ) + P(x =2) + .....+ P(x =10) + P(x =11)
= 12C0 * 0.25^0 * (1-0.25)^12 + 12C1 * 0.25^1 * (1-0.25)^11 + 12C2 * 0.25^2 * (1-0.25)^10 + 12C3 * 0.25^3 * (1-0.25)^9 + ............+ 12C10 * 0.25^10 * (1-0.25)^2 + 12C11 * 0.25^11 * (1-0.25)^1

= 1
  

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