Comparing Two Population Parameters

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Chapter 13

• Comparing Two Population Parameters

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13.1 Comparing Two Means

• Which of two drugs, Lipitor or Pravachol, help
  lower bad cholesterol more? An experiment was
  designed which used 4000 people with heart
  disease as subjects. They were randomly assigned
  to one of two treatment groups. At the end of
  the study researchers compared the mean bad
  cholesterol levels in both groups. For the
  Pravachol subjects, the mean was 95 milligrams
  per deciliter, for the Lipitor subjects, it was
  62 milligrams per deciliter. Is this difference
  statistically significant? Can Lipitor claim
  that it does a better job?

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• That question, and some of the most common
  questions in statistics involve two populations
  or two treatments. These are two-sample
  problems.
• Goal of inference is to compare the responses to
  two treatments OR compare the characteristics of
  two populations
• We have a separate sample from each treatment or
  each population.
• The responses of each group are independent of
  those in the other group.

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• Conditions for Comparing two Means
• SRS We must have two SRSs from two distinct
  populations. This allows us to generalize our
  findings. We measure the same variable from both
  samples.
• Normality Both populations are Normally
  distributed. In practice, it is enough that the
  distributions have similar shapes and that the
  data have no strong outliers. The mean and
  standard deviations of the populations are
  unknown.
• Independence The samples are independent. That
  is, one sample has no influence on the other.
  Paired observations violate independence. When
  sampling without replacement from two distinct
  populations, each population must be at least 10
  times as large as the corresponding sample.

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• Here is the notation that will be used to
  describe the two populations and samples.

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• There are four unknown parameters.
• We want to compare the two population means in
  one of two ways
• Give a confidence interval for µ1 - µ2
• Test the hypothesis of no difference H0 µ1
  µ2
• We start both by looking at the sample
  differences x-bar1 x-bar2.

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• Ex. 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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• The subjects in part of the experiment were 21
  healthy black men. A randomly chosen group of 10
  of the men received calcium supplements for 12
  weeks. The control group of 11 men received a
  placebo that looked identical. The experiment
  was double-blind. The response variable is the
  decrease in systolic blood pressure after 12
  weeks. Group 1 is the calcium group and Group2
  is the placebo group.

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• Here are the data for the calcium group
• 7 -4 18 17 -3 -5 1 10 11 -2
• Here are the data for the placebo group
• -1 12 -1 -3 3 -5 5 2 -11
• -1 -3

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• Step 1

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• Step 2 Two-sample t-Test

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• The Sampling Distribution of x-bar1 x-bar2
• The mean is µ1 - µ2, or it is an unbiased
  estimator. For our purposes, in all cases µ1 -
  µ2 0.
• The variance of the difference is the sum of the
  variances because the samples are independent.
• If the two populations are both Normal, then
  x-bar1 x-bar2 is also Normal.

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• Using this information, we can find the two
  sample z statistic

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• However, rarely do we know both population
  standard deviations. So, we must replace s with
  the estimate, s, the sample standard deviation of
  both populations.
• This gives the Two-sample t Statistic

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• The Two-sample t statistic does not have exactly
  a t distribution, but it is close enough for
  approximation.
• We need to know the degrees of freedom, which
  leaves us with two options.
• Use technology calculators and computers will
  use a messy formula to find the exact df.
• Estimate use the lesser df from n1 1 and n2
  1.
• Option 2 will give a conservative estimate of any
  p-value or critical t used in CI.

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• Step 3

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• Step 4

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• Now, construct and interpret a 90 confidence
  interval for the mean advantage of calcium over a
  placebo.

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• Robustness
• Two-sample procedures are more robust than
  one-sample when it comes to skewness.
• When the sizes of the two samples are equal and
  the distributions have similar shape, two-sample
  procedures are accurate with samples as small as
  n1 n2 5.
• Use the same guidelines as one-sample but replace
  n with n1 n2.
• In general, when planning a study, choose equal
  sample sizes if you can.

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• Approximation to Degrees of Freedom

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• Ex. Poisoning by the pesticide DDT causes
  convulsions in humans and other mammals.
  Researchers seek to understand how the
  convulsions are caused. In a randomized
  comparative experiment, they compared 6 white
  rats poisoned with DDT and a control group of 6
  unpoisoned rats. Electrical measurements of
  nerve activity are the main clue to the nature of
  DDT poisoning. When a nerve is stimulated, its
  electrical response shows a sharp spike followed
  by a much smaller second spike. The experiment
  found that the second spike is larger in rats fed
  DDT than in normal rats. This finding helps
  biologists understand how DDT poisoning works.
  The researchers measured the height (or
  amplitude) of the second spike as a percent of
  the first spike when a nerve in the rats leg was
  stimulated. For the poisoned rats the results
  were
• 12.207 16.869 25.050 22.429 8.456
  20.589
• The control data were
• 11.074 9.686 12.064 9.351 8.182
  6.642.
• Use a test to show that there is a significant
  difference in the two groups.

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• Step 1

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• Step 2

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• Step 3

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• Step 4