According to a study done by a university? student, the probability a randomly s

Question

According to a study done by a university? student, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267.

Suppose you sit on a bench in a mall and observe? people's habits as they sneeze.

?(a) What is the probability that among 10 randomly observed individuals exactly 4 do not cover their mouth when? sneezing?

?(b) What is the probability that among 10 randomly observed individuals fewer than 5 do not cover their mouth when? sneezing?

?(c) Would you be surprised? if, after observing 10 ?individuals, fewer than half covered their mouth when? sneezing? Why?

?(a) The probability that exactly 4 individuals do not cover their mouth is ________.

Explanation / Answer

X ~ Binomial (n,p)

Where, n = 10, p = 0.267

Binomial probability distribution is

P(X) = nCx px (1-p)n-x

a)

P( X = 4) = 10C4 0.2674 0.7336

= 0.1655

b)

P( X < 5) = P( X <= 4)

= P( X = 0) + P( X = 1) + P( X = 2) + P( X = 3) + P( X = 4)  

= 10C0 0.2670 0.73310 +10C1 0.2671 0.7339 +10C2 0.2672 0.7338 +10C3 0.2673 0.7337+ 10C4 0.2674 0.7336

= 0.9004

c)

P(individual covered their mouth when sneezing) = 1 - 0.267 = 0.733

Therefore,

P( Fewer than half covered theire mouth when sneezing) = P( X < 5)

= P( X <= 4)

= P( X = 0) + P( X = 1) + P( X = 2) + P( X = 3) + P( X = 4)  

= 10C0 0.7330 0.26710 +10C1 0.7331 0.2679 +10C2 0.7332 0.2678 +10C3 0.7333 0.2677+ 10C4 0.7334 0.2676

= 0.0272

Since probability is less than 0.05 (that is smaller) we would be surprised if after observing 10

individuals, fewer than half covered their mouth when sneezing.

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