Statistics 270

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Statistics 270 Lecture 25
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Cautions about Z-Tests

• Data must be a random sample
• Outliers can distort results
• Shape of the population distribution matters
  (large and small samples?)
• For significance tests, there is a difference
  between practical and statistical significance

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Types of Errors in a Significance Test

• Suppose that the null hypothesis is true, we
  could collect a sample that suggests that we
  reject H0
• Suppose that H0 is not true, we could fail to
  reject the null hypothesis

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Inference About the Population Mean

• To make inference about the population mean, m,
  we have used the z-test
• Key feature s is known
• Most often, s is unknown!
• s must be estimated from the data
• Use to estimate s

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Inference About the Population Mean

• To make probability statements about the the
  sample mean, we have used the Z-statistic when s
  is known
• When s is unknown, we use the one-sample
  t-statistic with (n-1) degrees of freedom

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Inference About the Population Mean

• Standard error
• Degrees of freedom

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One Sample t-Test for a Population Mean

• Data random sample x1, x2, , xn
• Mean, m, is unknown
• Standard deviation is unknown
• For testing the hypothesis H0 mm0
• Test Statistic
• Degrees of freedom t has a t-distribution with
  n-1 degrees of freedom

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One Sample t-Test for a Population Mean

• Computing p-value depends on the alternate
  hypothesis
• P-values are exact if the population distribution
  is normal and approximately correct for large
  samples in other cases

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One Sample t-Test for a Population Mean

• Rule of Thumb
• For nlt15, use t-test if data appear approximately
  normal
• For n 15, can use t-test when no outliers
• For large n (say n 40), can safely use t-test
• Note always require random sample!

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Example

• Composition of earths atmosphere has changed
  over time
• Gas bubbles in ancient amber are examined to
  study the nature of the atmosphere long ago
• Measurements on specimens of amber from the
  Cretaceous period (75-95 million years ago) give
  the following percentages of nitrogen

63.4 65.0 64.4
63.3 54.8 64.5
60.8 49.1 51.0
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Example

• Assume the data are a random sample from the
  Cretaceous period
• To see if there is a difference with todays
  78.1 nitrogen, conduct a hypothesis test using
  these data

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Example

• Hypotheses
• Test Statistic
• P-value
• Conclusion