Chapter 7 Estimation

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Chapter 7 Estimation

• Instructor Xijin Ge
• SDSU Dept. of Math/Stat

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Sampling situation how many hours do SDSU
students spend in social networking sites?
Population11,400 students in SDSU
Random Sample 1
100 students
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Evaluating estimators by repeated sampling and
estimation
Population11,400 students in SDSU
Random Sample 1
Sample mean
Random Sample 2
Random Sample 3
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An Experiment

• Suppose r.v. X is a number we got from rolling an
  honest die
• p.d.f.
• This is the exact value of the mean that we
  derived theoretically.

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Estimation of the mean

• Pretend that we dont know the exact value of the
  mean and want to evaluate it through experiments.
• Rolling die 30 times and calculate average
• 56 students did the experiments

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Histogram of all point estimates
x scan(data.txt) hist(x,xlimc(2.5,4.5))
Unbiased centered at the right spot.
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Defining Unbiased estimator

• An estimator is an unbiased estimator for a
  parameter if and only if

To test if the statistic is an unbiased estimator
we need to repeat the sampling and estimation
many, many times. If the average of the
estimated values approaches the true/theoretical
value, then it is unbiased.
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Theorem 7.1.1
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• Are unbiased estimations accurate?
• We sampled 100 students and
• Can we guarantee that the true mean is close to
  1.5hr?

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Desirable properties of point estimator

• Unbiased, and
• Small variance for large sample sizes.

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• 10 minutes group quiz
• (total 5 points)
• Give answer (1 point)
• Prove it (2 points)
• Discuss it (2 points)

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For larger sample size, the difference between
repeated estimation is smaller.
Population11,400 students in SDSU
Random Sample 1
Sample mean
Random Sample 2
Random Sample 3
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Discussions on Theorem 7.1.2

• Sample means based on small sample may differ
  significantly from actual population mean.
• Sample mean based on a large sample can be
  expected to lie reasonable close to actual
  population mean.
• Standard deviation of
  This is called standard error of the mean.

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Th. 7.1.3. Estimation of variance
Proof in Appendix C. Note S is not an unbiased
estimator of s.
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Distribution of X
Properties of Moment Generating Functions
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X is normally distributed !
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Expected value of your learning outcome

• What is an unbiased estimator?
• Sample mean is an unbiased estimator of
  population mean.
• Variance of sample mean decrease linearly with
  the increase of sample size.
• Unbiased estimation of variance is S2
• X is normally distributed!