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

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 16 randomly observed individuals exactly 8 do not cover their mouth when sneezing?

(b) What is the probability that among 16 randomly observed individuals fewer than 4 do not cover their mouth when sneezing?

(c) Would you be surprised if, after observing 16 individuals, fewer than half covered their mouth when sneezing? Why?

Explanation / Answer

a) P(X = 8) = 16C8 * (0.267)^8 * (0.733)^8 = 0.028

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

                  = 16C0 * (0.267)^0 * (0.733)^16 + 16C1 * (0.267)^1 * (0.733)^15 + 16C2 * (0.267)^2 * (0.733)^14 + 16C3 * (0.267)^3 * (0.733)^13 = 0.346

c) P(X < 8) = P(X = 0) + P(X = 1 ) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5 ) + P(X = 6) + P(X = 7)

                  = 16C0 * (0.733)^0 * (0.267)^16 + 16C1 * (0.733)^1 * (0.267)^15 + 16C2 * (0.733)^2 * (0.267)^14 + 16C3 * (0.733)^3 * (0.267)^13 +16C4 * (0.733)^4 * (0.267)^12 + 16C5 * (0.733)^5 * (0.267)^11 + 16C6 * (0.733)^6 * (0.267)^10 + 16C7 * (0.733)^7 * (0.267)^9 = 0.011

As the probability is less than 0.05, so it is unusual.

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