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 . Suppose you sit on a bench in a mall and observe​ people's habits as they sneeze. ​(a) What is the probability that among randomly observed individuals exactly do not cover their mouth when​ sneezing? ​(b) What is the probability that among randomly observed individuals fewer than do not cover their mouth when​ sneezing? ​(c) Would you be surprised​ if, after observing ​individuals, fewer than half covered their mouth when​ sneezing? Why? ​(a) The probability that exactly individuals do not cover their mouth is nothing. ​(Round to four decimal places as​ needed.) ​(b) The probability that fewer than individuals do not cover their mouth is nothing. ​(Round to four decimal places as​ needed.) ​(​c) Fewer than half of individuals covering their mouth ▼ would would not be surprising because the probability of observing fewer than half covering their mouth when sneezing is nothing​, which ▼ is is not an unusual event. ​(Round to four decimal places as​ needed.)

Complete Question

The complete question is shown on the first uploaded image

Answer:

a

 [tex]P(X = 8) =  0.0037[/tex]

b

 [tex]P(X <  5) =  0.805[/tex]

c

 [tex]P(X > 6) =  0.0206[/tex]

I would be surprised because the value is very small , less the 0.05

Step-by-step explanation:

From the question we are told that

The probability a randomly selected individual will not cover his or her mouth when sneezing is [tex]p = 0.267[/tex]

Generally data collected from this study follows  binomial  distribution because the number of trials is  finite , there are only two outcomes, (covering  , and  not covering mouth when sneezing ) , the trial are independent

Hence for a randomly selected variable  X we have that  

   [tex]X \ \ \~ \ \ { B ( p , n )}[/tex]

The probability distribution function for binomial  distribution is  

    [tex]P(X = x ) =  ^nC_x *  p^x *  (1 -p) ^{n-x}[/tex]

Considering question a

Generally the  the probability that among 12 randomly observed individuals exactly 8 do not cover their mouth when​ sneezing is mathematically represented as

     [tex]P(X = 8) =  ^{12} C_8 *  (0.267)^8 *  (1- 0.267)^{12-8}[/tex]

Here C denotes  combination

So

     [tex]P(X = 8) =  495  *  0.000025828 * 0.28867947[/tex]

    [tex]P(X = 8) =  0.0037[/tex]

Considering question b

Generally the probability that among 12 randomly observed individuals fewer than 5 do not cover their mouth when​ sneezing is mathematically represented as

     [tex]P(X <  5 ) =[P(X = 0 ) + \cdots + P(X = 4)][/tex]

=>   [tex]P(X <  5 ) =[ ^{12} C_0 *  (0.267)^0 *  (1- 0.267)^{12-0} + \cdots +  ^{12} C_4 *  (0.267)^4 *  (1- 0.267)^{12-4} ][/tex]

=> [tex]P(X <  5 )  =  0.02406 +  0.10516 + 0.21067 + 0.25580 + 0.20964[/tex]

=>  [tex]P(X <  5) =  0.805[/tex]

Considering question c

Generally the probability that fewer than half(6) covered their mouth when​ sneezing(i.e the probability the greater than half do not cover their mouth when sneezing) is mathematically represented as

      [tex]P(X > 6) =  1 - p(X \le  6)[/tex]

=>    [tex]P(X > 6) = 1 - [P(X = 0) + \cdots + P(X =6)][/tex]

=>    [tex] P(X > 6)=1 - [^{12} C_0 *  (0.267)^0 *  (1- 0.267)^{12-0}+ \cdots + ^{12} C_4 *  (0.267)^6 *  (1- 0.267)^{12-6} ][/tex]

=>    [tex] P(X > 6)= 1 - [0.02406 + \cdots + 0.0519 ][/tex]  

=>    [tex]P(X > 6) =  0.0206[/tex]

I would be surprised because the value is very small , less the 0.05

(a )randomly observed individuals exactly do not cover their mouth when​ sneezing

[tex]P[x=8]=0.0037[/tex]

(b) randomly observed individuals fewer than do not cover their mouth when​ sneezing

[tex]P[x < 5]=0.805[/tex]

(c)  fewer than half covered their mouth when​ sneezing

[tex]P(x > 6)=0.0206[/tex]

It is given in the question that

The probability a randomly selected individual will not cover his or her mouth when sneezing is

[tex]P=0.267[/tex]

Generally, data collected from this study follows  binomial  distribution because the number of trials is  finite, there are only two outcomes, (covering, and  not covering mouth when sneezing ), the trial is independent

The probability distribution function for binomial  distribution is  

[tex]P(x=8)=n C_X\times p^x\times (1-p)^{n-x}[/tex]

Generally, the probability that among 12 randomly observed individuals exactly 8 does not cover their mouth when​ sneezing is mathematically represented as

[tex]P(x=8)=12c_8\times (0.267)^8\times (1-0.267)^{12-8}[/tex]  

[tex]p(x=8)=495\times 0.000025828\times 0.2886794[/tex]

[tex]p(x=8)=0.0037[/tex]

Generally, the probability that among 12 randomly observed individuals fewer than 5 do not cover their mouth when​ sneezing is mathematically represented as

[tex]p(x < 5)=[p(x=0)+......+p9x=4)][/tex]

[tex]p(x < 5)=[12c_0\times(0.267)^0\times(1-0.267)^{12-0}+........+12c_4\times(0.267)^4\times (1-0.267)^{12-4}[/tex]

[tex]p(x < 5)=0.02406+0.10516+0.21067+0.25580+0.20964[/tex]

[tex]p(x < 5)=0.805[/tex]

Generally the probability that fewer than half(6) covered their mouth when​ sneezing(i.e the probability the greater than half do not cover their mouth when sneezing) is mathematically represented as

[tex]p(x > 6)=1-p(x\leq 6)[/tex]

[tex]p(x > 6)=1-[p(x=6)+......+p(x=6)][/tex]

[tex]p(x > 6)=1-[12c_0(0.267)^0\times(1-0.267)^{12-0}+....+12c_4\times(0.267)^6\times(1-0.267)[/tex]

[tex]p(x > 6)=1-[0.02406+.....+0.0519][/tex]

[tex]p(x > 6)=0.0206[/tex]

Thus

(a )randomly observed individuals exactly do not cover their mouth when​ sneezing

[tex]P[x=8]=0.0037[/tex]

(b) randomly observed individuals fewer than do not cover their mouth when​ sneezing

[tex]P[x < 5]=0.805[/tex]

(c)  fewer than half covered their mouth when​ sneezing

[tex]P(x > 6)=0.0206[/tex]

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