Inference for One Sample

1
Inference for One Sample

• Essential Ideas and Techniques

2
Recall 2 Types of Inference

• Estimation
• Goal Provide a specific approximation for a
  parameter
• Result Numerical Interval
• Tests of Hypotheses
• Goal Judge whether a parameter might equal a
  particular value
• Result Yes or No

3
Confidence Intervals

• Idea an estimate that reflects the uncertainty
  inherent in sampling
• Approach find sample mean, and then bracket it

4
Confidence Interval Procedure

• Select a LEVEL OF CONFIDENCE
• Determine the MARGIN OF SAMPLING ERROR
• The margin depends on the Sampling Distribution
  AND the level of confidence
• Create the interval
• Sample mean Margin of Error

5
An Illustrative Example

• Problem 6.16-- by hand
• Confidence Level .95
• Determine Critical Value of T
• Compute xbar and s
• n12, mean443.33, std dev14.28
• Develop interval
• Now recall the Minitab solution
• Graphical Summary
• T-distribution command

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Use of Confidence Intervals

• Interpretation
• Problem 6.17
• We attach probability to the interval NOT to m.

• We can also use an intervalto test a hypothesis
• Would this sample evidence support the idea that
  m 450?
• How about m 455?

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Tests of Hypotheses

• A Hypothesis is an informed guess about the
  value of a parameter
• Examples
• m 450
• m gt 98.6
• s1 s2

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Each Test Has 2 Hypotheses

• NULL Hypothesis Ho
• Assertion about what m equals
• The Reigning Champion
• ALTERNATIVE Hypothesis Ha
• Represents the complement of null
• The Challenger
• Generally, the alternative hypothesis represents
  the reason for our investigation

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An illustrative example

• Problem 6.19, parts a b
• Minitab solution
• Understanding the results

10
The Logic of the Approach

• Assume the null is true
• Rules of Evidence
• Anticipate the likely extent of sampling error,
  given a sample of size n.
• Decision Rule
• If sample result is consistent with anticipated
  sampling error, persist in assuming null is true
• If sample result is inconsistent with anticipated
  error, reject the null

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Interpreting the Result

• Compute test statistic (z-score)
• EITHER
• Compare Test Statistic to a Critical z
• Compare Area beyond test statistic to
  Significance Level (a)
• The latter approach is known as the P-Value
  method
• Preferable when you have a computer

12
Other Issues

• Begin with Alternative Hypothesis
• Alternatives can be
• Two sided m ?a value
• One-sided
• m gt a value or
• m lt a value
• Use t-test unless Population is non-normal AND
  small sample