1. Sampling Distributions and the Central Limit Theorem
CHAPTER 5
Sampling Distributions and
the Central Limit Theorem

2. Population Distribution of X
Suppose X ~ N(μ, σ), then…
X = Age of women in U.S. who have given birth X
Density     x4 x5 x1 x2  x3
… etc….
σ = 1.5
Most individual ages are in the neighborhood of μ, but there are occasional outliers in the tails of the distribution. x x x x x
μ = 25.4

3. Population Distribution of X
Suppose X ~ N(μ, σ), then…
Sample,
n = 400
Sample,
n = 400
X = Age of women in U.S. who have given birth
Sample,
n = 400
Sample,
n = 400 X
Density
Sample,
n = 400
… etc….
How are these values distributed?
σ = 1.5
μ = 25.4

4. Population Distribution of X
Sampling Distribution of
for any sample size n.
Suppose X ~ N(μ, σ), then…
Suppose X ~ N(μ, σ), then…
X = Age of women in U.S. who have given birth
Density
μ = X
Density
μ =
σ = 1.5
“standard error”
… etc….
How are these values distributed?
The vast majority of sample mean ages are extremely close to μ, i.e., extremely small variability.
μ = 25.4

5. Population Distribution of X
Sampling Distribution of
Suppose X ~ N(μ, σ), then…
Suppose X ~ N(μ, σ), then…  
for large sample size n.
for any sample size n.
X = Age of women in U.S. who have given birth
Density
μ = X
Density
μ =
σ = 2.4
Suppose X ~ N(μ, σ), then…
“standard error”
… etc….
How are these values distributed?
The vast majority of sample mean ages are extremely close to μ, i.e., extremely small variability.
μ = 25.4

6. Population Distribution of X
Sampling Distribution of
X ~ Anything with finite μ and σ
Suppose X  N(μ, σ), then… 
for large sample size n.
for any sample size n.
X = Age of women in U.S. who have given birth
Density
μ = X
Density
μ =
σ = 2.4
Suppose X ~ N(μ, σ), then…
“standard error”
… etc….
How are these values distributed?
The vast majority of sample mean ages are extremely close to μ, i.e., extremely small variability.
μ = 25.4