ECON 400
September 17, 2019

Estimation of Means and Proportions
Parameters are the numerical measures of a population that describe
probability distributions. We have seen many of these such as
p
,
n
,
r
,
N
, λ, θ
etc.
Sample statistics such as ¯
x
,
s
can also be used to make inference
about their population counterparts,
μ, σ
.
The parameters
p
, λ, θ
that we have studied are population
parameters. Their sample counterparts are ˆ
p
,
ˆ
λ,
ˆ
θ
.
Statistics
September 17, 2019
2 / 48

Estimation of Means and Proportions
The sample statistics are used to estimate their corresponding
population parameters. However, this is not a perfect measure, of
course. We also need to decide what the best sample statistic is for
each population parameter, and we need some way to decide
between them.
In some cases for example, the sample statistics of the sample
average, median, or mode may be he best choice to estimate
μ
.
Statistics
September 17, 2019
3 / 48

Estimation of Means and Proportions
No one sample statistic will always be closest to the population
parameter (the truth).
Let’s consider an example where we know the population parameter,
the truth. Consider flipping a coin 4 times. Say we are interested in
the number of heads. We know from our knowledge of probability
theory that the expected number of heads is two of the four tosses.
However our results will vary with finite tosses. Either the mean,
median, or mode may be closest.
This is easy to discount as we know the population parameter.
What if we were discussing the amount of time a person in a survey
says they spend on the phone each day. We certainly don’t know
what the result would be if we surveyed every possible person, the
population. Then our choice is not so obvious.
Statistics
September 17, 2019
4 / 48

Sampling Distribution
Sample statistics are variables, with their own probability
distributions. They are known as
Sampling Distributions
.
We can see the behavior of a sample statistic by taking many
samples, and producing a histogram of the result, approximating the
probability density function of ¯
x
/the sample median/the sample
mode/etc.
Consider ¯
x
from a distribution with mean
μ
= 10 and
σ
= 2. If we
take many samples from this distribution and calculate the means,
we can plot the sample means into a histogram to see how close ¯
x
came to the population parameter
μ
= 10
Statistics
September 17, 2019
5 / 48

Sampling Distribution
The following histograms were generated by telling the computer to
draw a sample of 50 from a distribution with mean 10 and standard
deviation 2, 1000 times. The 1000 sample means, medians, and
modes were used to construct the following histograms. Tell me
which sample statistic you think performed better.
Statistics
September 17, 2019
6 / 48

Estimation of Means and Proportions
Statistics
September 17, 2019
7 / 48

Estimation of Means and Proportions
The mean and median both performed well. The average of each of
the mean, median and modes were: 10.0082, 10.0046, and 5.5204.

#### You've reached the end of your free preview.

Want to read all 48 pages?

- Spring '08
- turchi