STATISTICAL INFERENCE PART V

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STATISTICAL INFERENCEPART V

• CONFIDENCE INTERVALS

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INTERVAL ESTIMATION

• Point estimation of ? The inference is a guess
  of a single value as the value of ?. No accuracy
  associated with it.
• Interval estimation for ? Specify an interval
  in which the unknown parameter, ?, is likely to
  lie. It contains measure of accuracy through
  variance.

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INTERVAL ESTIMATION

• An interval with random end points is called a
  random interval. E.g.,

is a random interval that contains the true value
of ? with probability 0.95.
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Interpretation

• ( )
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• µ (unknown, but true value)

90 CI ? Expect 9 out of 10 intervals to cover
the true µ
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INTERVAL ESTIMATION

• An interval (l(x1,x2,,xn), u(x1,x2,,xn)) is
  called a 100? confidence interval (CI) for ? if
• where 0lt?lt1.
• The observed values l(x1,x2,,xn) is a lower
  confidence limit and u(x1,x2,,xn) is an upper
  confidence limit. The probability ? is called the
  confidence coefficient or the confidence level.

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INTERVAL ESTIMATION

• If Pr(l(x1,x2,,xn)??) ?, then l(x1,x2,,xn) is
  called a one-sided lower 100? confidence limit
  for ? .
• If Pr(?? u(x1,x2,,xn)) ?, then u(x1,x2,,xn) is
  called a one-sided upper 100? confidence limit
  for ? .

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METHODS OF FINDING PIVOTAL QUANTITIES

• PIVOTAL QUANTITY METHOD
• If Qq(x1,x2,,xn) is a r.v. that is a
  function of only X1,,Xn and ?, and if its
  distribution does not depend on ? or any other
  unknown parameter (nuisance parameters), then Q
  is called a pivotal quantity.

nuisance parameters parameters that are not of
direct interest
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PIVOTAL QUANTITY METHOD

• Theorem Let X1,X2,,Xn be a r.s. from a
  distribution with pdf f(x?) for ??? and assume
  that an MLE (or ss) of ? exists
• If ? is a location parameter, then Q ?? is a
  pivotal quantity.
• If ? is a scale parameter, then Q /? is a
  pivotal quantity.
• If ?1 and ?2 are location and scale parameters
  respectively, then

are PQs for ?1 and ? 2.
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Note

• Example If ?1 and ?2 are location and scale
  parameters respectively, then
• is NOT a pivotal quantity for ?1
• because it is a function of ?2
• A pivotal quantity for ?1 should be a function of
  only ?1 and Xs, and its distribution should be
  free of ?1 and ?2 .

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Example

• X1,,Xn be a r.s. from Exp(?). Then,
• is SS for ?, and ? is a scale parameter.
• S/? is a pivotal quantity.
• So is 2S/?, and using this might be more
  convenient since this has a distribution of
  ?²(2n) which has tabulated percentiles.

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CONSTRUCTION OF CI USING PIVOTAL QUANTITIES

• If Q is a PQ for a parameter ? and if percentiles
  of Q say q1 and q2 are available such that
• Prq1? Q ?q2?,
• Then for an observed sample x1,x2,,xn a 100?
  confidence region for ? is the set of ??? that
  satisfy q1? q(x1,x2,,xn?)?q2.

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EXAMPLE

• Let X1,X2,,Xn be a r.s. of Exp(?), ?gt0. Find a
  100? CI for ? . Interpret the result.

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EXAMPLE

• Let X1,X2,,Xn be a r.s. of N(?,?2). Find a 100?
  CI for ? and ?2 . Interpret the results.

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APPROXIMATE CI USING CLT

• Let X1,X2,,Xn be a r.s.
• By CLT,

Non-normal random sample
The approximate 100(1-?) random interval for µ
The approximate 100(1 -?) CI for µ
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APPROXIMATE CI USING CLT

• Usually, ? is unknown. So, the approximate
  100(1??) CI for ?

Non-normal random sample

• When the sample size n 30, t?/2,n-1N(0,1).

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Graphical Demonstration of the Confidence
Interval for m
1 - a
Lower confidence limit
Upper confidence limit
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Inference About the Population Mean when ? is
Unknown

• The Student t Distribution

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Effect of the Degrees of Freedom on the t Density
Function

The degrees of freedom (a function of the
sample size) determines how spread the
distribution is compared to the normal
distribution.
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Finding t-scores Under a t-Distribution (t-tables)

.05
t
1.812
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EXAMPLE

• A new breakfast cereal is test-marked for 1 month
  at stores of a large supermarket chain. The
  result for a sample of 16 stores indicate average
  sales of 1200 with a sample standard deviation
  of 180. Set up 99 confidence interval estimate
  of the true average sales of this new breakfast
  cereal. Assume normality.

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ANSWER

• 99 CI for ?
• (1067.3985, 1332.6015)
• With 99 confidence, the limits 1067.3985 and
  1332.6015 cover the true average sales of the new
  breakfast cereal.

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Checking the required conditions

• We need to check that the population is normally
  distributed, or at least not extremely nonnormal.
• Look at the sample histograms, Q-Q plots
• There are statistical methods to test for
  normality