How do you set up fewer than half of 16 individuals covering their mouth would be surprising because the probability of observing fewer than half covering their mouth when sneezing is 0.011 8 which is an unusual event

Check Explanation

Step-by-step explanation:

To set up the probability that fewer than half of 16 cover their mouth while sneezing.

It will be modelled as a binomial distribution problem

Binomial distribution function is represented by

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = number of people observed = 16

x = Number of successes required = less than 8

p = probability of success = probability that one covers his/her nose = 1 - 0.267 = 0.733

q = probability of failure = probability that one doesn't cover his/her nose = 0.267

So, probability of less than 8 people covering their nose is P(X < 8)

And it is given mathematically as a sum of probabilities from 0 person covering his/her nose to 7 people covering their nose

P(X < 8) = Σ ⁿCₓ pˣ qⁿ⁻ˣ (summing from x = 0 to x = 7 with n constant at 16)

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)

P(X < 8) = 0.0 + 0.0 + 0.0 + 0.00000772941 + 0.000068964 + 0.0004543875 + 0.00228697031 + 0.00896923079

P(X < 8) = 0.0117872825 = 0.0118 as given in the question.