Question

To illustrate the law of large numbers, which we mentioned in connection with the frequency interpretation of probability, suppose that among all adults living in a large city w know that there are as many men as women. Using the normal-curve approximation, find the probabilities that in a random sample of adults living in this city the proportion of men will be any where from 0.49 to 0.51 when the number of persons in the sample is

a) 100; b) 1000; c)10000.

answer is a)0.2358 b)0.4908 c)0.9556. Please show the formula and the detail steps . Thank you!

Answer 1

a)

for normal distribution z score =(p̂-p)/σp
here population proportion=     p=	0.500
sample size       =n=	100
std error of proportion=σp=√(p*(1-p)/n)=	0.0500

probability =

P(0.49<X<0.51)

=

P(-0.2<Z<0.2)=

0.5793-0.4207=

0.1586

b)

std error of proportion=σp=√(p*(1-p)/n)=

0.0158

probability =

P(0.49<X<0.51)

=

P(-0.63<Z<0.63)=

0.7365-0.2635=

0.4730

c)

std error of proportion=σp=√(p*(1-p)/n)=

0.0050

probability =

P(0.49<X<0.51)

=

P(-2<Z<2)=

0.9772-0.0228=

0.9544