Estimation Bias, Standard Error and Sampling Distribution

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Estimation Bias, Standard Error and Sampling
Distribution
Topic 9
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From sample to population

• Inductive (inferential) statistical methods

Make inference about a population based on
information from a sample derived from that
population
Population
inductive statistical methods
sample
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Statistical Concepts of Sampling

• Suppose we want to estimate the mean birthweight
  of Malay male live births in Singapore, 1992
• Due to logistical constraints, we decide to take
  a random sample of 50 live births from the
  records of all Malay male live births for that
  year

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Sampling from Target Population

Target population
random sample of 50 Malay male live births in
Singapore, 1992
All Malay male live births in Singapore, 1992
Suppose sample mean 3.55 kg sample SD (S)
0.92 kg What can we say about the population mean?
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Statistical Modeling

• Assume the population values follow a normal or
  some other appropriate distribution. This means a
  relative frequency histogram of the population
  values will look like a normal or that
  appropriate distribution.
• Assume we have a random sample, i.e., we sample
  n (50 in example) values independently from the
  population

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Notation
Sample data
Assume
are independent and each is
distributed according to say a normal distribution
Population parameters
Population mean mean of the normal population
Population variance variance of the normal
population
Population standard deviation
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Statistical Inference
Two general areas (a) Statistical
Estimation i.e. estimating population parameters
based on sample statistics
(b) Hypothesis Testing i.e. testing certain
assumptions about the population
Also called Test of Statistical Significance
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Statistical Estimation

• There are two ways by which a population
  parameter can be estimated from a sample
• (1) Point estimate
• (2) Interval estimate

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Point Estimate

• Estimate the population parameter by a
• single value
• Sample mean population mean
• Sample median population median
• Sample variance population variance
• Sample SD population SD
• Sample proportion population proportion

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Point Estimate

• If the average birthweight for a random sample of
  Malay male births was 3.55 kg and we use it to
  estimate m, the mean birthweight of all Malay
  male births in the population, we would be making
  a point estimate for m

• Poor practice to report just the point estimate
  because people cannot judge how good the estimate
  is
• Should also report the accuracy of the estimate.
• Remember that the quality of an estimator is
  judged by its performance over REPEATED SAMPLING
  although we have just one sample in hand.

Inference for population parameter should make
allowance for sampling error
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Accuracy of statistical estimation

• Two types of error
• (a) Sampling error or fluctuation
• random error or fluctuation that is due
  entirely to chance in the process of sampling.
  Minimizing the sampling error maximizes the
  precision of a statistical estimate.

(b) Systematic error or bias Non-random
error/bias which is either a property of the
estimator itself or due to bias in the sampling
or measurement process. Minimizing the
systematic error maximizes the validity of a
statistical estimate. Systematic errors can be
minimized by making efforts to reduce measurement
bias (eg non-random sampling, non-response and
non-coverage, untruthful answers, unreliable
calibration, errors with data recording and
coding etc)
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Unbiased estimation of the mean
i.e., the sample mean equals the population
mean when averaged over repeated samples
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Hypothetical results of repeated sampling

• Unbiasedness means the sample mean equals the
  population mean when averaged over repeated
  samples
• However, there is fluctuation from sample to
  sample
• Variance ?

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Standard Error (SE) of an estimator

• The SE of an estimator (e.g., the sample mean) is
  just the standard deviation (SD) of the
  estimator. It measures the variability of the
  estimator under repeated sampling
• SE is just a special case of SD
• The reason why the standard deviation of an
  estimator is called standard error is because it
  is a measure the magnitude of the estimation
  error due to sampling fluctuation

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Standard Deviation vs Standard Error

• The population standard deviation (SD) measures
  the amount of variation among the individual
  measurements that make up the population and can
  be estimated from a sample using the sample
  standard deviation.
• The standard error (e.g. of the sample mean), on
  the other hand, measures how much the value of
  the estimator changes from sample to sample under
  repeated sampling.
• As we take only 1 sample rather that repeated
  samples in practice, it seems impossible at first
  to estimate standard error which is defined with
  reference to repeated sampling.
• Fortunately, the standard error of the sample
  mean is a function of the population SD. As the
  latter is estimable from a single sample, so is
  the standard error.

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Estimated standard error of the sample mean

• Let denote the population SD
• It was shown earlier that
• SE SD(sample mean) / , where n is
  the sample size
• Since can be estimated by the sample standard
  deviation S, we can estimate the standard error
  by SE S/

Note that SE decreases with n at the rate 1/
, i.e., the precision of the sample mean improves
as sample size increases
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Knowing the mean and standard error of an
estimator still doesnt tell us the whole story
The whole story is told by the sampling
distribution since that helps in calculating the
probabilities
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Sampling distribution of the sample mean

• The distribution of the sample mean under
  repeated sampling from the population

• Distribution of the sample mean rather than
  individual measurements
• In practice, we take only one sample, not
  repeated samples and so the sampling distribution
  is unobserved but fortunately it can often be
  derived theoretically

Demo http//www.ruf.rice.edu/lane/stat_sim/inde
x.html
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Exact result when sampling from a normal
population

• If the population is normal with mean and
  variance , then the sample mean based on a
  random sample of size n is also normal with mean
  and variance
• Note how we can derive theoretically the
  distribution of the sample mean under repeated
  sampling without actually drawing repeated
  samples
• This is important because we usually only have
  one sample at our disposal in practice

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Topic 10 Interval Estimate

• Provides an estimate of the population parameter
  by defining an interval or range of plausible
  values within which the population parameter
  could be found with a given confidence.
• This interval is called a confidence interval.
• The sampling distribution is used in constructing
  confidence intervals.

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Confidence interval for the mean of a normal
population
Fact With probability 0.95, a normally
distributed variable is within 1.96 standard
deviations from its mean.
Now

• It follows that the sample mean must be within
  1.96 standard errors from the population mean
  with probability 0.95.
• Equivalently, the population mean is within 1.96
  standard errors from the sample mean.

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We call
a 95 confidence interval for the population mean.
If is unknown, replace it by the sample SD
and replace 1.96 by the upper 2.5-percentile of a
t-distribution with n-1 degrees of freedom to
yield
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as a 95 confidence interval for the population
mean
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The t densities

• t densities are symmetric and similar in
  appearance to N(0,1) density but with heavier
  tails
• Tables for t distributions are widely available
• As d.f. increases, t distribution converges to
  standard normal distribution

Demo http//www.isds.duke.edu/sites/java.html
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95 confidence interval for the population mean
Birthweight data revisited

• n 100, Sample mean 3.55 kg, S 0.92 kg
• SE .92/sqrt(50) 0.13 kg
• d.f. 49, upper 2.5-percentile of t 2.01
• 95 C.I. for the mean Malay male birthweight is
• 3.55 /- 2.01 (0.13) (3.29 kg, 3.81 kg)

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The meaning of confidence interval
Under repeated sampling,
will contain the true mean 95 of the times.
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Demo http//www.isds.duke.edu/sites/java.html