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

0.2670.267.

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

​(a) What is the probability that among

1818

randomly observed individuals exactly

55

do not cover their mouth when​ sneezing?

​(b) What is the probability that among

1818

randomly observed individuals fewer than

33

do not cover their mouth when​ sneezing?

​(c) Would you be surprised​ if, after observing

1818

​individuals, fewer than half covered their mouth when​ sneezing? Why?

Answer 1

Answer:

Given that:

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

The probability that a randomly selected person does not cover his/her mouth while sneezing. That p = 0.267.
Let X be the number of persons in a mall who will sneeze.

a) the probability that among 18 randomly observed individuals exactly 5 do not cover their mouth when? sneezing.
P(X = 55) = ¹⁸C_⁵(0.267)^5 (1-0.267)^18-5

= 8568*0.001356*0.017634
= 0.2050
Hence, the required probability is, 0.2050

b) The probability that among 18 randomly observed individuals fewer than 3 do not cover their mouth when? sneezing.
P(X < 3) = P(X = 0) +P(X = 1) +P(X = 2)
= ¹⁸C₀ (0.267)^0 (1-0.267)^18-0 + ¹⁸C₁ (0.267)^1 (1-0.267)^18-1 + ¹⁸C₂ (0.267)^2 (1-0.267)^18-2
= 0.0037+ 0.0244+ 0.0757
= 0.1038
Hence, the required probability is, 0.1038

c) Would you be surprised? if, after observing 18?individuals, fewer than half covered their mouth when? sneezing.
P(X < 9) = P(X = 0) +P(X = 1) +...... + P(X = 6) +P(X = 7)+P(X = 8)
= 0.0037+ 0.0244+ 0.0757+ 0.1471+0.2010+ 0.2050 + 0.1618+ 0.1010 + 0.7098
= 1.6295
Hence, the required probability is, 1.6295

It is observe that the fewer than half of the 18 persons do not cover their mouth we would doubt about the proposed probability 'p' .and would comment crudely that may be the proportion of persons those who do not cover their mouth is more than 0.267 .