Confidence Intervals

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Confidence Intervals

• Ginger Holmes Rowell, Ph. D.
• April 10, 2002

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Estimation Inference

• Point Estimators
• one value estimates a population parameter
• Examples
• Sample Mean estimates _______
• Proportion of sample estimates ____________

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Estimation Inference

• Interval Estimators
• a range of values estimates a population
  parameter
• Example Confidence Intervals
• For estimating
• Population Means
• Population Proportions

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Confidence Intervals

• Two Features
• Interval Estimate of the population parameter
• Level of Confidence associated with the estimate

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Confidence Intervals

• Confidence Interval
• Point Estimator /- Margin of Error

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Confidence Intervals

• Example for Means
• Consider the average number of students in a
  teachers class
• Form you hypothesis
• What do you think?

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Example for Means

• Collect the data
• Class Size Data
• 29, 27, 30, 28, 28, 30, 29, 21, 22, 29, 26, 31,
  28, 25, 34, 23, 24, 29, 34, 27, 30, 25, 25, 30,
  30, 27, 29, 29, 29, 28
• Sample Size 30

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Class Size Data

• Sample Size 30
• Sample Mean 27.9 students
• Sample Standard Deviation 3 people
• Data are collected from a Normal population with
  standard deviation of 3 students

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The Empirical Rule

• IF the data are symmetrical and mound shaped,
  then approximately
• 68 of the data fall within 1 standard deviation
  from the mean
• 95 of the data fall within 2 standard deviations
  from the mean
• Almost 100 of the data fall within 3 standard
  deviation from the mean

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The Empirical Rule
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Forming a 95 CI

• An approximation for 95 confidence in an
  estimate of the average class size

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Forming a 95 CI

• sample mean /- 2SE
• SE Standard Error
• Standard Deviation of the Mean Standard
  Deviation/Square root of sample size

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Forming a 95 CI

• sample mean /- 2SE
• 27.9 /- 2(3/sqrt(30))
• 27.9 /- 1.1
• (27.9 - 1.1, 27.9 1.1)
• (26.8, 29.0)

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Interpretation

• We are 95 confident that the true population
  average class size is between 26.8 and 29
  students.

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General Interpretation 95 CI

• If we sample repeatedly, then each time we would
  get a different sample mean and thus the
  confidence interval would change.

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General Interpretation 95 CI

• If we did this 100 times, we would expect 95 of
  the intervals to contain the true population mean
  and 5 of the intervals would not.

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General Interpretation 95 CI

• The only problem is that you dont know if your
  confidence interval from your sample is one of
  the 95 that is correct, or if it is one of the 5
  that is wrong.

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Using Your TI-83 for CI

• Press STATgtTESTSgtZinterval
• Input Stat
• sigma 3
• x-bar 27.9
• n 30
• C-Level 95
• select Calculate

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Using Your TI-83 for CI

• (26,826, 28.974)
• x-bar27.9
• n30

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Using Your TI-83 for CI

• Now find the following confidence intervals for
  the same example
• 99
• 90
• 80

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CIs for population average class size

• 99 - (26.489, 29.311)
• 95 - (26,826, 28.974)
• 90 - (26.999, 28.801)
• 80 - (27.198, 28.602)
• Conclusions about confidence level and width of
  confidence interval?

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What is the effect of sample size on CI width?

• What is your intuition?
• Try a 95 CI with a sample size of 100
• (27.312, 28.488)
• The 95 CI with n30 is
• (26,826, 28.974)

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Applications

• Time to cook a pizza
• Time to cook microwave popcorn
• Very important in the medical profession
• ...

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Questions

• ???