Answer: a) 0.0125 b) 0.1454 c) Mean = 5.7, standard deviation = 1.73
Step-by-step explanation:
Let x = Number of children say that chocolate chip cookie is their favorite kind of cookie.
x follows binomial distribution.
Binomila distribution formula:
[tex]P(X=x)=\ ^nC_xp^x(1-p)^{n-x}[/tex] , where p = probability in each trial, n = sample size, x= number of successes
Sample size : n= 12, p=0.475
a)
[tex]P(x\geg10)= P(x=10)+P(x=11)+P(x=12)\\\\=\ ^{12}C_{10}(0.475)^{10}(1-0.475)^{2}+^{12}C_{11}(0.475)^{11}(1-0.475)^{1}+^{12}C_{12}(0.475)^{12}(1-0.475)^{0}\\\\=\dfrac{12!}{2!10!}(0.475)^{10}(0.525)^{2}+(12)(0.475)^{11}(0.525)^{1}+(1)(0.475)^{12}\\\\\approx0.0125[/tex]
b)
[tex]P(x=4)= ^{12}C_{4}(0.475)^4(0.525)^{8}\\\\=\dfrac{12!}{4!8!}(0.000293798)\\\\=0.1454[/tex]
c) Mean = np
= (12)(0.475) =5.7
Standard deviation = [tex]\sqrt{np(1-p)}[/tex]
[tex]=\sqrt{12\times0.475\times0.525}\\\\=\sqrt{2.9925}\approx1.73[/tex]
