Download Sampling Theory

Sampling Theory
Dennis Sun
Data 301
The Need for Theory
When we were testing whether a coin was fair, the null hypothesis
completely specified the box model:
0 1 .
This allowed us to simulate from the box model and see whether
the observed data is consistent with those simulations.
The Need for Theory
Now suppose we have an instrument that measures the height of a
mountain. The measurements are centered around the true
height, but with a standard deviation of 10 feet.
We make 15 measurements of the height of Mt. Everest, and find
that the average is 29,023 feet. The official height is 29,029 feet.
Has Mt. Everest gotten shorter?
The null hypothesis is that Mt. Everest has not gotten shorter.
What does this tell us about the box?
? ? ... ?
µ = 29029
σ = 10
We don’t actually know the tickets in the box, so we can’t simulate
from the box model. We need theory!
Central Limit Theorem
Mean of 100 draws from 0 1
Central Limit Theorem
Mean of 100 draws from 0 1
If you take the mean of n draws from any box, the distribution of
the means will be approximately Normal(µ, √σn ) when n is large.
Central Limit Theorem
• The Central Limit Theorem says that the mean of 100 draws
from 0 1 is approximately Normal(.5, √.5
).
100
• Instead of simulating from the box model, we can simulate
from the normal distribution.
• Using the normal approximation, the probability of observing
more than 60 heads is 2.3%. (Exact answer is 2.8%.)
Central Limit Theorem
The benefit of the Central Limit Theorem is that you just need to
know the mean and SD of the box. You don’t need to know the
tickets in the box!
This allows us to answer the Everest problem, where we don’t
know the exact composition of the box, but we do know its mean
and SD under the null hypothesis.