Paired Samples and Blocks

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Chapter 25

• Paired Samples and Blocks

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Independent vs. Dependent Samples

• We would like to determine if students taking an
  ACT prep course will score better than students
  not taking the course. A random sample of 25
  students was chosen who took the course and a
  random sample of another 25 students was chosen
  who did not take the course. At the end of the
  prep course, both groups were given the ACT.

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Independent vs. Dependent Samples

• Group 1 score on ACT from students taking prep
  course
• Group 2 score on ACT from students NOT taking
  prep course
• Observations taken from two independent samples

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Independent vs. Dependent Samples

• We would like to determine if students can
  improve their ACT score by taking a prep course.
  A random sample of 25 students was chosen. They
  first took the ACT test. Then they spent 6 weeks
  taking the prep course. At the end of the 6
  weeks, they took the ACT test again.

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Independent vs. Dependent Samples

• Group 1 score on ACT before prep course
• Group 2 score on ACT after prep course
• Two observations taken for each subject

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Important Difference

• If the values come from
• two independent samples
• use inference for (µ1 µ2) difference in means
  for two groups
• dependent samples (e.g. values collected twice
  from same subject)
• use inference for µd mean difference between
  two values
• The second situation is called Matched Pairs

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Inference for µd

• d pairwise difference
• Confidence Interval
• Hypothesis Test (degrees of freedom n-1)

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Pairwise Differences

• ACT Prep Course Does the course improve scores?

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Matched Pairs Hypothesis Test Example

• Measure effectiveness of exercise program in
  lowering blood cholesterol levels
• SRS of men from a population
• Measure cholesterol before program starts
• Go through exercise program for 12 weeks
• Measure cholesterol after program ends
• Is the exercise program effective?

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Matched Pairs Hypothesis Test Example - Data
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Matched Pairs Hypothesis TestExample

• Two observations from each subject
• Post value depends on pre value
• Observations are dependent
• Do not have independent samples
• Cannot use inference for µ1 - µ2
• Violates independent samples assumption

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Matched Pairs Hypothesis Test Example - Data

• Look at pre-program levels minus post-program
  levels

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Matched Pairs

• Because it is the differences we are interested
  in, we will treat them as the data and ignore the
  original two groups.
• A matched pairs t-test is just a one-sample
  t-test (from Chapter 23) for the mean of the
  pairwise differences.

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Matched Pairs Hypothesis Test Example

• Let µd mean difference in cholesterol levels in
  population
• Sample mean difference is
• Sample standard deviation of differences is sd
  17.99
• n number of differences 10

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Matched Pairs Hypothesis Test Example

• If exercise program has no effect, mean
  difference will be 0
• If exercise program is effective, mean difference
  will be positive
• Step 1
• HO µd 0
• HA µd gt 0

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Matched Pairs Hypothesis Test Example

• Step 2
• Assumptions
• Random sample
• Nearly Normal Population
• Test statistic

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Matched Pairs Hypothesis Test Example

• Step 3
• P-value
• P(t9 gt 1.42)
• between 0.05 and 0.10

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Matched Pairs Hypothesis Test Example

• Step 4
• Decision Since p-value gt 0.05 a, we will fail
  to reject the null hypothesis.
• Conclusion There is no evidence that the
  exercise program reduced the mean cholesterol
  level of men in this population.

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Matched Pairs Confidence Interval Example

• For a random sample of 12 European cities, the
  average high temperatures in January and July are
  given on the following slide.
• Find a 90 confidence interval for the mean
  temperature difference between summer and winter
  in Europe. Assume the Nearly Normal assumption
  is satisfied.

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Matched Pairs Confidence Interval Example
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Matched Pairs Confidence Interval Example

• Check assumptions
• Random
• ok
• Nearly Normal
• ok

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Matched Pairs Confidence Interval Example