1. The sampling distribution is the distribution of all possible values that can be assumed by some statistic computed from samples of the same size randomly drawn from the same population. 2. To construct a sampling distribution, all possible samples of a given size are drawn from the population and the statistic is computed for each sample. The distinct observed values and their frequencies are listed. 3. According to the central limit theorem, the sampling distribution of the sample mean will be approximately normally distributed for large sample sizes, regardless of the population distribution.

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SAMPLING DISTRIBUTIONS
And
t DISTRIBUTION

2. Definition
 The distribution of all possible values that can be
assumed by some statistic, computed from samples
of the same size randomly drawn from the same
population, is called the sampling distribution of
that statistic.
SAMPLING DISTRIBUTIONS

3. Sampling Distribution : Construction
To Construct a Sampling Distribution We Proceed as
Follows:
 From a finite population of size N, randomly draw all
possible samples of size n.
 Compute the statistic of interest for each sample.
 List in one column the different distinct observed
values of the statistic, and in another column list the
corresponding frequency of occurrence of each
distinct observed value of the statistic.

4. Properties of Sampling Distribution
 The distribution of will be normal.
 The mean, of the distribution of will be equal
to the mean of the population from which the samples
were drawn.
 The variance, of the distribution of will be
equal to the variance of the population divided by the
sample size.
x
x ,µ x
x
2
σ x

5. The Central Limit Theorem
 Given a population of any nonnormal functional
form with a mean µ and finite variance σ2
, the
sampling distribution of , computed from samples
of size n from this population, will have mean µ and
variance σ2
/ n and will be approximately normally
distributed when the sample size is large.
x
A Mathematical Formulation of the Central
Limit Theorem
x
/ n
− µ
σ

6. The Sampling Distribution of : A Summary
1. Sampling is from a normally distributed population
with a known population variance:
(a)
(b)
(c) The sampling distribution of is normal
x
xµ = µ
x / nσ = σ
x

7. 2. Sampling is from a nonnormally distributed population
with a known population variance:
(a)
(b)
otherwise
(c) The sampling distribution of is approximately
normal
xµ = µ
x / n , when n / N .05σ = σ ≤
x
N n
( / n) ,
N 1
−
σ = σ
−
x

8. Properties of the t Distribution
 The t distribution has the following properties.
1. It has a mean of 0.
2. It is symmetrical about the mean.
3. In general, it has a variance greater than 1, but
the variance approaches 1 as the sample size
becomes large. For df > 2, the variance of the t
distribution is df / (df − 2), where df is the degrees
of freedom. Alternatively, since here df
= n − 1 for n > 3, we may write the variance of the t
distribution as (n − 1) / (n − 3).
4. The variable t ranges from − ∞ to + ∞.

9. Figure. The t distribution for different degrees-of-freedom values

10. 5. The t distribution is really a family of distributions,
since there is a different distribution for each sample
value of n − 1, the divisor used in computing s2
. We
recall that n − 1 is referred to as degrees of freedom.
Figure shows t distributions corresponding to several
degrees -of-freedom values.
6. Compared to the normal distribution, the t
distribution, the t distribution is less peaked in the
center and has thicker tails. Figure compares the t
distribution with the normal.
7. The t distribution approaches the normal
distribution as n − 1 approaches infinity.

11. Figure. Comparison of normal distribution and t distribution