Two Independent Means

1
Two Independent Means

• Unit 8

2
Sampling Considerations

• One sample or two?
• If two samples, paired or independent?
• Is the response variable quantitative or
  categorical?
• Am I interested in the mean difference?

This chapter ? two independent samples ?
quantitative response ? interest in mean
difference
3
One sample
SRS from one population
Comparisons made to an external reference
population
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Paired Sample
Two samples with each observation in sample 1
matched to a unique observation in sample 2
Just like a one-sample problem except inferences
directed toward within-pair differences DELTA
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Independent sample inference
Independent samples from two populations
No matching or pairing
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What type of sampling method?

• Measure vitamin content in loaves of bread and
  see if the average meets national standards.
• Compare vitamin content of bread immediately
  after baking versus 3 days later (same loaves are
  used on day one and 3 days later)
• Compare vitamin content of bread immediately
  after baking versus loaves that have been on
  shelf for 3 days

• 1 single sample
• 2 paired samples
• 3 independent samples

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Illustrative example independent samples
Goal compare response variable in two groups

• Fasting cholesterol (mg/dl)
• Group 1 (type A personality) 233, 291, 312, 250,
  246, 197, 268, 224, 239, 239, 254, 276, 234, 181,
  248, 252, 202, 218, 212, 325
• Group 2 (type B personality) 344, 185, 263, 246,
  224, 212, 188, 250, 148, 169, 226, 175, 242, 252,
  153, 183, 137, 202, 194, 213

8
Data setup for independent samples

• Two columns
• Response variable in one column
• Explanatory variable in other column

9
Side-by-side boxplots
Compare locations, spreads, and shapes

• Interpretation
• Different locations (group 1 gt group 2)
• Different spreads (group 1 lt group 2)
• Shape fairly symmetrical (but both with outside
  values)

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Summary statistics by group
If no major departures from Normality, report
means and standard deviations (and sample sizes)
Take time to look at your results.
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Notation for independent samples
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Sampling distribution of mean difference
The sampling distribution of the mean difference
is key to inference
FIGURE DRAWN ON BOARD
The SDM difference tends to be Normal with
expectation µ1 - µ2 and standard deviation SE
(SE discussed next slide)
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Pooled Standard Error
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Confidence interval for µ1 µ2
(1-alpha)100 confidence interval for µ1 µ2
Illustrative example (Cholesterol in type A and B
men)
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Comparison of CI formulas
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Independent t test
Pooled t statistic

• A. H0 µ1 µ2 vs. H1 µ1 gt µ2 or H1 µ1 lt
  µ2 or H1 µ1 ? µ2
• B. Independent t statistic
• C. P-value use t table or software utility to
  convert tstat to P- value
• D. Significance level

Illustrative example
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SPSS output
These are the pooled (equal variance) statistics
calculated in HS 167
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Conditions necessary for t procedures

• Validity assumptions
• good information (no information bias)
• good sample (no selection bias)
• good comparison (no confounding no lurking
  variables)
• Distributional assumptions
• Sampling independence
• Normality
• Equal variance

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Sample size requirements for confidence intervals
This will restrict the margin of error to no
bigger than plus or minus d
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Sample size requirement for CI

• Suppose, you have a variable with s 15

Sample size requirements increases when you need
precision
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Sample size for significance test

• Goal to conduct a significance test with
  adequate power to detect a difference worth
  detecting
• The difference worth detecting is a difference
  difference worth finding.
• In a study of an anti-hypertensivesfor instance,
  a drop of 10 mm Hg might be worth detecting,
  while a drop of 1 mm Hg might not be worth
  detecting.
• In a study on weight loss, a drop of 5 pounds
  might be meaningful in a population of runway
  models, but may be meaningless in a morbidly
  obese population.

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Determinants of sample size requirements

• Difference worth detecting (?)
• Standard deviation of data (s)
• Type I error rate (a)
• We consider only a .05 two-sided
• Power of test (we consider on 80 power)

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Sample size requirements for test

• Approx. sample size needed for 80 power at alpha
  .05 (two-sided) to detect a difference of ?

Illustrative example Suppose ? 25 and s 45