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Presentation 3
More about the Sampling
Distribution of the Sample Mean
and
introduction to the
t-distribution
1
Brief Review of Sampling Distributions
Sampling Distribution of Sample Proportion:
 Each member of the population has a trait of interest
with probability p (population proportion).
 Suppose a random sample of size n is obtained form the
population.
 The sample proportion p-hat is a logical estimator of p,
number of element in the sample with trait
pˆ 
total sample size

If the sample is large enough, np and n(1-p) >5, then
pˆ
 p (1  p ) 
~ N p,
Z 
n


approx .
pˆ  p approx .
~ N (0,1)
p (1  p )
n
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Sampling Distribution of Sample Mean



Let X be a random variable and the statistic be the
sample mean of X in a random sample of size n.
We examine the sampling distribution of in the
following three scenarios:
1.
When X is a normal random variable, E(X)=µ and
s.d.(X)= σ are both known.
2.
When X is not a normal random variable, E(X)=µ
and s.d.(X)= σ are both known and the sample size
is large, n ≥ 30.
3.
When X is a normal random variable, E(X)=µ is
known, s.d.(X)= σ is unknown and the sample size is
large, n ≥ 30.
So far we have seen the first two cases.
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
Case 1:
E (X )  

2



X 

2


Var ( X )     X ~ N   ,  
~ N (0, 1)
n   n


X normal r.v.

Case 2:
E (X )  


Var ( X )   2   X

n  30


 2 
X   approx
 
~ N   ,
~ N (0, 1)
n   n

approx
Case 3:
E (X )  

X 

sample s.d.(X )  s  
~ t n 1, where tn-1 denotes the
s n

t-distribution with n -1
X normal r.v.

degrees of freedom.
4
Properties of the t-distribution






There are infinitely many t-distributions, each
characterized by one parameter, the degrees of
freedom.
The degrees of freedom are positive integers, e.g. t1,
t2 , t3,…, t10,…
Random variables with t-distribution are continuous.
The density curve of a t - distribution is symmetric,
bell-shaped and centered at zero (similar to the
standard normal curve).
There are tables for the probabilities related with a t –
random variable. In will see how to use them later in
the course.
As the degrees of freedom increase, the variance of
the t -random variable decreases, i.e. the density
curve is less spread, and actually it approaches the
standard normal density.
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Properties of the t-distribution
Z~N(0,1)
t10
t3
t1
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