1Please type the assumptions required for a binomial normal

1)Please type the assumptions required for a binomial normal approximation using inequalities and the following symbols (n,p,q) where n is the sample size, p is the probability parameter from the binomial experiment, and q=(1-p).

(write both answers together separated by a single space).

3) The probability of having a girl is 0.5, Using normal approximation, what is the probability that a family of five will have exactly one boy (assume for the family of five they have 3 children, give answer to three decimal places)? _____________

(hint, rewrite the question as a probability statement that has a non-zero probability)

(hint, normal distribution is continuous, binomial is discrete)

Find the exact binomial probability _____________

Solution

1) let X~Bin(n,p) so E[X]=np   V[X]=npq   where q=1-p

so for normal approximation

X would follow approximately a normal distribution with mean np and variance npq if the sample size i.e, n is very large i.e, when n tensed to infinity. in general for practical purpose for n>=25 X approximately follows a normal distribution.

2.)N(25,1²) point of inflection- \"(25-1,25+1)\"=\"(24,26)\"

N(100,2²)   point of inflection- \"(100-2,100+2)\"=\"(98,102)\"

N(200,4²)      point of inflection- \"(200-4,200+4)\"=\"(196,204)\"

3) let X be the random variable denoting the number of boys in a family with 3 children.

now probability of having a boy=1-probability of having a girl=1-0.5=0.5

so X~Bin(3,0.5)

exact binomial probability=P[X=1]=³C₁0.5³=0.375 [answer]

now E[X]=3*0.5=1.5 V[X]=3*0.5*0.5=0.75

using normal approximation we have X~N(mean=1.5,variance=0.75)

so P[X=1]=0389974 [answer]