Section 11.2 Comparing Two Means

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Section 11.2Comparing Two Means

• AP Statistics
• www.toddfadoir.com/apstats

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Comparing Two Means

• Very useful to compare two populations
• Two populations equates to two distributions
• Easier math to work with one distribution
• Distribution of the difference of means

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Conditions for Comparing Two Means

• We have two SRSs, from two distinct populations.
• The samples are independent. That is, one sample
  has no influence on the other. Matching violates
  independence, for example.
• We measure the same variable for both samples.
• Both populations are normally distributed. The
  means and standard deviations of the populations
  are unknown.

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Example

• Does increasing the amount of calcium in our diet
  reduce blood pressure? Examination of a large
  sample of people revealed a relationship between
  calcium intake and blood pressure. The
  relationship was strongest for black men. Such
  observational studies do not establish causation.
  Researchers therefore designed a randomized
  comparative experiment.

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Example

• The subjects in part of the experiment were 21
  healthy black men. A randomly chosen group of 10
  of the men received a calcium supplement for 12
  weeks.
• The group of 11 men received a placebo pill that
  looked identical.
• The experiment was double-blind.
• The response variable is the decrease in systolic
  (heart contracted) blood pressure for a subject
  after 12 weeks, in millimeters of mercury. An
  increase appears as a negative response.

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Example

• Group 1 (Calcium) results
• 7, -4, 18, 17, -3, -5, 1, 10, 11, -2
• n10, x-bar5.000, s8.743
• Group 2 (Placebo) results
• -1, 12, -1, -3, 3, -5, 5, 2, -11, -1, -3
• n11, x-bar-0.273, s5.901

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Inference Tool Box

• Step 1 Identify the populations and the
  parameters of interest you want to draw
  conclusions about. State hypothesis in words and
  symbols.
• Pop1 Black Men on Calcium
• Pop2 Black Men on Placebo
• Parameters of interest mean differences in blood
  pressure
• H0 µ1 µ2 (There is no difference in the blood
  pressure changes)
• Ha µ1gt µ2 (The men taking calcium see a larger
  decrease in blood pressure)

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Inference Tool Box

• Step 2 Choose the appropriate inference
  procedure, and verify the conditions for using
  the selected procedure.
• Test? Because we dont know the population
  standard deviation, well use a t test. Since
  were not comparing a person with himself, we
  have two sample.
• Independent? SRSs, therefore independent.
• Normal? Use back-to-back stemplots to check for
  normality. The book says no departures from
  normality

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Inference Tool Box

• Step 3 Compute the test statistic and and the
  P-value.
• Test? Because we dont know the population
  standard deviation, well use a t test. Since
  were not comparing a person with himself, we
  have two sample.
• Independent? SRSs, therefore independent.
• Normal? Use back-to-back stemplots to check for
  normality. The book says no departures from
  normality

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Example

• Step 3 Compute the test statistic and the
  P-value.
• Remember the P-value is also dependent on the
  degrees of freedom.

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Exercises

• 11.37-11.49 odd
• 11.63-11.73 odd