Confidence Intervals

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Confidence Intervals

• Hansel Burley

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Review

• Underlying dist. Distribution of all possible
  outcomes of an event
• Normal dist.a common underlying dist.
• Sampling dist. Of the mean-underlying dist for
  means of a given sample size
• Three important distributions
• scores in the pop
• scores in the random sample
• sampling distribution

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• Central limit theorem
• Standard error of the mean
• As sample size increases
• Simple random sampling
• Systematic sampling
• Cluster sampling
• Stratified Random sample
• Hypothesis testing
• Hypothesis

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One sample case for the mean

• 1. State the hypothesis
• 2. Set the criterion for rejection hypothesis
• Errors (Type I or Type II)
• Level of significance (alpha level)
• Region of rejection
• 3. Compute the test statistic
• 4. Decide about the hypothesis

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• Non directional test region of rejection is
  split
• Directional test region of rejection is either
  greater or less than the hypothesis
• These tests use the normal distribution as the
  underlying distribution when sigma is known.
• When sigma is unknown, use Student t
  distributions
• Test statisticstatistic-parameter/standard error
  of the statistic.
• Standard error of the statisticsigma/square root
  of n
• Standard error of the statistics/square root of
  n

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Confidence Intervals

• Statistical estimation inferences about pop.
• Point estimate a single value represents the
  best estimate of the parameter
• Interval estimation a range of values are
  probable (tenable) for the parameter. You are
  confident that this range of values includes
  the parameter.

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• Usually sigma is unknown
• Therefore use sample variance
• When sigma is unknown, use t distribution rather
  than the normal distribution
• Use estimated standard error of the mean

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Testing many hypotheses

• If you computed sample means of all possible
  samples and constructed the 95 confidence
  interval, 95 would contain the parameter and 5
  would not.
• Use mean, standard error and critical value

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Statistical precision

• Narrower the confidence interval, the more
  precise the estimate.
• Increasing sample size increases precision
• As level of confidence increases, the width of
  the interval increases.

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• If the mean is 43
• Std.Dev10
• N400
• For 95 confidence interval
• 43 or- (1.96)(10/square root of n)
• The parameter will fall between 42.02 and 43.98.