47.5% of children say that chocolate chip cookie is their favorite kind of cookie. If you randomly select 12 children find the probability that: a) at least 10 of them say that chocolate chip cookie is their favorite cookie. B) four of them say that chocolate chip cookie is their favorite cookie. C) Find the mean and standard deviation for this sample.

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]

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