Lecture 20: Point and Interval Estimation

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Lecture 20 Point and Interval Estimation

• Professor Aurobindo Ghosh
• E-mail ghosh_at_galton.econ.uiuc.edu

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Introduction

• Statistical inference is the process by which we
  acquire information about populations from
  samples.
• There are two procedures for making inferences
• Estimation.
• Hypotheses testing.

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Concepts of Estimation

• The objective of estimation is to determine the
  value of a population parameter on the basis of a
  sample statistic.
• There are two types of estimators
• Point Estimator
• Interval estimator

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Point Estimator

• A point estimator draws inference about a
  population by estimating the value of an unknown
  parameter using a single value or a point.

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• Point Estimator
• A point estimator draws inference about a
  population by estimating the value of an unknown
  parameter using a single value or a point.

Parameter
Population distribution
?
Sample distribution
Point estimator
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Interval Estimator

• An interval estimator draws inferences about a
  population by estimating the value of an unknown
  parameter using an interval.
• The interval estimator is affected by the sample
  size.

Interval estimator
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• Selecting the right sample statistic to estimate
  a parameter value depends on the characteristics
  of the statistics.

• Estimators desirable characteristics
• Unbiasedness An unbiased estimator is one whose
  expected value is equal to the parameter it
  estimates.
• Consistency An unbiased estimator is said to be
  consistent if the difference between the
  estimator and the parameter grows smaller as the
  sample size increases.
• Relative efficiency For two unbiased estimators,
  the one with a smaller variance is said to be
  relatively efficient.

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Estimating the Population Mean when the
Population Standard Deviation is Known

• How is an interval estimator produced from a
  sampling distribution?
• To estimate m, a sample of size n is drawn from
  the population, and its mean is calculated.
• Under certain conditions, is normally
  distributed (or approximately normally
  distributed.), thus

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• We know that

• This leads to the relationship

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1 - a
Upper confidence limit
Lower confidence limit
See simulation results demonstrating this point
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• The confidence interval are correct most, but
  not all, of the time.

UCL
LCL
Not all the confidence intervals cover the real
expected value of 100.
100
0
The selected confidence level is 90, and 10 out
of 100 intervals do not cover the real m.
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• Four commonly used confidence levels

za/2
The mean values obtained in repeated draws of
samples of size 100 result in interval
estimators of the form sample mean - .28,
Sample mean .28 90 of which cover the real
mean of the distribution.
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• Recalculate the confidence interval for 95
  confidence level.
• Solution

• The width of the 90 confidence interval
  2(.28) .56
• The width of the 95 confidence interval
  2(.34) .68
• Because the 95 confidence interval is wider,
  it is more likely to include the value of m.

.95
.90
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• Example 9.1
• The number and the types of television programs
  and commercials targeted at children is affected
  by the amount of time children watch TV.
• A survey was conducted among 100 North American
  children, in which they were asked to record the
  number of hours they watched TV per week.
• The population standard deviation of TV watch was
  known to be s 8.0
• Estimate the watch time with 95 confidence
  level.

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• Solution

• The parameter to be estimated is m, the mean time
  of TV watch per week per child (of all American
  Children).
• We need to compute the interval estimator for m.
• From the data provided in file XM09-01, the
  sample mean is

Since 1 - a .95, a .05. Thus a/2 .025.
Z.025 1.96
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• Interpreting the interval estimate
• It is wrong to state that the interval
  estimator is an interval for which there is 1 - a
  chance that the population mean lies between the
  LCL and the UCL.
• This is so because the m is a parameter, not a
  random variable.