1
Hypothesis Testing
Estimation
We use sample statistics to estimate population parameters.
We want our sample
estimates to be good in the sense that the sample numbers are close to the population
values.
Q1:
is the sample statistic equal to the population parameter on average?
If the answer to this question is "yes," the statistic is said to be an
unbiased
estimator of
the parameter.
If the answer is "no," the statistic is a
biased
estimator of the parameter.
Q2:
what is the average distance of the statistic from the parameter?
If the statistic is
unbiased, this is the standard deviation of the statistic, also known as the
standard error
.
Other things being, equal, we want the average statistic to equal the parameter, and we
want the average distance from the parameter to the statistic to be small.
Sampling distribution
The sampling distribution is what we get when we do the following:
1.
Take a sample of size N (a given number) from a population (with replacement).
2.
Compute the statistic (parameter estimate,
X
,
s, s
2
, r
) and record it.
3.
Repeat steps 1 and 2 a lot (infinitely).
4.
The resulting distribution, that is, the distribution of the statistic that comes from the
repeated samples is called a
sampling distribution
.
Estimating
the Mean
Example:
estimating mean height of USF students.
75
72
69
66
63
60
40
30
20
10
0
Height in Inches
Frequency
Frequency Distribution and Sampling Distribution
Population of height
Sampling Distribution of Mean Height
(N=100)
Points to notice:
1. The mean of the sampling distribution is close to the mean of the population.
2. The standard deviation of the sampling distribution is much smaller than the standard
deviation of the population.
The relation between the two can be expressed:
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σ
X
N
=
where
X
is the standard deviation of the sampling distribution of the mean,
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 Fall '08
 Staff

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