Answer: a) 0.3011
b) 0.3526
c) 0.6455
Step-by-step explanation:
Given : The proportion of adults would pay more for environmentally friendly products : p= 0.21
Sample size : n= 10
Let x be a binomial variable that denotes the number of adults would pay more for environmentally friendly products.
Using binomial distribution, [tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]
a) The probability that the number of adults who would pay more for environmentally friendly products is exactly 2 will be :-
[tex]P(x=2)=^{10}C_2(0.21)^2(1-0.21)^{10-2}\\\\=\dfrac{10!}{2!(10-2)!}(0.21)^2(0.79)^8\approx0.3011[/tex]
The probability that the number of adults who would pay more for environmentally friendly products is exactly 2=0.3011
b) The probability that the number of adults who would pay more for environmentally friendly products is more than two will be :-
[tex]P(x>2)=1-P(x\leq2)=1-(P(x=0)+P(x=1)+P(x=2))\\\\=1-(^{10}C_0(0.21)^0(1-0.21)^{10}+^{10}C_1(0.21)^1(1-0.21)^{9}+0.3011)\\\\=1-((0.79)^{10}+(10)(0.21)(0.79)^9+0.3011)\\\\=1-(0.0946+0.2517+0.3011)\\\\=1-0.6474=0.3526[/tex]
The probability that the number of adults who would pay more for environmentally friendly products is more than two =0.3526
c) The probability that the number of adults who would pay more for environmentally friendly products is between two and five, inclusive will be :-
[tex]P(2\leq x\leq5)=P(x=2)+P(x=3)+P(x=4)+P(x=5)\\\\=0.3011+^{10}C_3(0.21)^3(1-0.21)^7}+^{10}C_4(0.21)^4(1-0.21)^{6}+^{10}C_5(0.21)^5(1-0.21)^{5})\\\\=0.3011+\dfrac{10!}{3!7!}(0.21)^3(0.79)^7+\dfrac{10!}{4!6!}(0.21)^4(0.79)^6+\dfrac{10!}{5!5!}(0.21)^5(0.79)^5\\\\=0.3011+0.2134+0.0993+0.0317=0.6455[/tex]
The probability that the number of adults who would pay more for environmentally friendly products is between two and five, inclusive =0.6455
