Question

compute the sampling distribution for three tosses of an unfair coin. Assume the random variable x=1 if the coin is tails and x=0 if it is head. The probability of coin is head facing upward is 0.47.

a) compute the mean and standard deviation of the population.

b) what is the sampling distribution of the sample mean

c) compute the mean and standard deviation of the sampling distribution

d) as n gets large does the distribution of sample mean approach a normal distribution? explain.

Answer 1

a)

	X	P(X)	x.P(x)	x^2.P(X=x)
Head	1	0.47	0.47	1
Tail	0	0.53	0	0
		1	0.47	1

Mean ,

Variance ,  

Standard deviation of X ,

b) Sample size , n=3

Sample space is given by

	X
H,H,H	3
H,H,T	2
H,T,H	2
T,H,H	2
H,T,T	1
T,H,T	1
T,T,H	1
T,T,T	0

Probability distribution of sample mean is

	X		P()
H,H,H	3	1	0.103823
H,H,T	2	0.666667	0.117077
H,T,H	2	0.666667	0.117077
T,H,H	2	0.666667	0.117077
H,T,T	1	0.333333	0.132023
T,H,T	1	0.333333	0.132023
T,T,H	1	0.333333	0.132023
T,T,T	0	0	0.148877

c) Mean of sampling distribution of sample mean

= 0.47

Standard deviation of sampling distribution of sample mean

=(0.3039-0.47^2)

=0.288

Calculation :

	X		P()	.P()	2P()
	3	1	0.103823	0.103823	0.103823
	2	0.666667	0.117077	0.078051	0.052034
	2	0.666667	0.117077	0.078051	0.052034
	2	0.666667	0.117077	0.078051	0.052034
	1	0.333333	0.132023	0.044008	0.014669
	1	0.333333	0.132023	0.044008	0.014669
	1	0.333333	0.132023	0.044008	0.014669
	0	0	0.148877	0	0
total			1	0.47	0.303933

d) Yes as n gets large , sampling distribution of sample mean follow Normal

We know that by Central Limit theorem , the sampling distribution of sample mean follow Normal even if population distribution is not Normal .

with

=0.47 ( population mean )

and

where