Download Sampling Distribution of Sample Mean

Chapter 7: Sampling Distributions
Consider a random process, a different sample of the same size drawn from the
same population to yield different values of the sample mean, x . The sample
mean, x is a random variable and possess a distribution, therefore, the
distribution of the sample mean is called sampling distribution of x .
Definition 7.1 :
A statistic is a function of the random variable in a random sample.
The probability distribution of a statistic is called its sampling distribution.
7.1 Sampling Distribution of Sample Mean
Definition 7.2 : Sampling Distribution of Sample Mean, x
The probability distribution of x is called its sampling distribution. It list the
various values that x can assume and the probability of each value of x .
In general, the probability distribution of a sample statistic is called its sampling
distribution.
Definition 7.3 : Mean and Standard Deviation of Sample Mean, x
The mean and standard deviation of the sampling distribution of x are called
the mean and standard deviation of x and denoted by  x and  x respectively.
The mean of the sampling distribution is equal to the mean of population. Thus,
x = 
whereas the standard deviation is given by,
x =

2
, V( x ) =
n
n
where  is the standard deviation of the population and n is the sample size.
7.1.1 Properties and Shape of The Sampling Distribution of The Sample
Mean, x
1. If the sample size, n  30, the sampling distribution of the sample mean is
normally distributed, where
2
x ~ N ( , )
n
Note : If the  unknown then it is estimated by s 2 .
2
2. If the sample size, n is considered small (<30), the sampling distribution of
the sample mean is normally distributed if the sample is from the normal
population and variance is known,
2
x ~ N ( , )
n
3. t distribution with n-1 degree of freedom if the sample is from the normal
population but the variance is unknown,
t
xu
2
s
n
~ t n 1
Z value for a value of x
The Z value for a value of x is calculated as
x
Z
x
7.2 Sampling Distribution of Sample Proportion
Population and Sample Proportion
The population and sample proportion are denoted by p and p̂ , respectively, are
calculated as,
X
x
p
and p̂ =
N
n
where
N = total number of elements in the population
X = number of elements in the population that possess a specific characteristic
n = total number of elements in the sample
x = number of elements in the sample that possess a specific characteristic
Definition 7.4 : Sampling Distribution of Sample Proportion, p̂ for Infinite
Population
The probability distribution of the sample proportion p̂ , is called its sampling
distribution. It gives various values that p̂ can assume and their probabilities.
For the large values of n (n  30), the sampling distribution is very closely
normally distributed.
Definition 7.5 : Mean of the Sample Proportion
The mean of the sample proportion, p̂ is denoted by  p̂ and is equal to the
population proportion, p. Thus,
 p̂ = p
Definition 7.6 : Standard Deviation of the Sample Proportion
 pˆ 
pq
n
where p is the population proportion, q= 1- p and n is the sample size.
 p(1  p) 
p̂ ~ N  p,

n


For a small values of n;
The population is binomial distributed X ~ B(n,p) and ;
X
P( p̂ = ) = P(X = x) = P( X  x) nCx p x qn x x = 0, 1, 2, ......, n
n
E ( X )  np V ( X )  npq
Where
Z value for a value of p̂
The Z value for a value of p̂ is calculated as
pˆ  p
Z
 pˆ