Question

(a) Using the rbinom or sample commands in R programming, simulate 50 independent aian (b) Repeat for 100 tosses, and then 500 tosses. For each of these calculate the proportion of heads and the standard deviation. (c) What do you observe? Is it because of the Law of Large Numbers or because of the Central Limit Theorem? Explain. (d) Now draw 500 samples of 500 tosses each. What is the mean proportion of heads? (e) Plot the sampling distribution for the proportion of heads. Is it standard normal? Explain. (f) What is the probability that the proportion of heads is greater than 52%? What is the probability that it is less than 48%?

Answer 1

run the followung comands on R software,

n=50;

x=rbinom(n,1,0.5) # note here i take 1 as a no. of trial.

l=length(x(x==1))

p=l/n # you will fond the proportion which is just near to 25 but not exact.

# for standard deviation. Run the following command

sd=sqrt(n*p*(1-p))

B--

# repeat these steps for n=100, n=50p you will find as n increase proportion approaches to .5

C-- it is due to law of large numbers as n increase x bar tends to polpn mean i.e .5

D-- run the following command on R

n=500;

m=500;

i=0

s[i]=0

for(i in 1:m){

s[i]=mean(rbinom(n,1,.5))

p=mean(s[i])

}

print(p)

E-- for sampling distribution plot

hist(x)

Any doubt comment