Political Science 30: Political Inquiry

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Political Science 30Political Inquiry
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The Magic of the Normal Curve

• Normal Curves (Essentials, pp. 127-130)
• The family of normal curves
• The rule of 68-95-99.7
• The Central Limit Theorem (Essentials, p. 127)
• Confidence Intervals (Essentials, pp. 130-134)
• Around a Mean
• Around a Proportion (Essentials, pp. 138-140)

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Normal Curves (or distributions)

• Normal curves are a special family of density
  curves, which are graphs that answer the
  question what proportion of my cases take on
  values that fall within a certain range?
• Many things in nature, such as sizes of animals
  and errors in astronomic calculations, happen to
  be normally distributed.

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Normal Curves
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Normal Curves

• What do all normal curves have in common?
• Symmetric
• Mean Median
• Bell-shaped, with most of their density in center
  and little in the tails

• How can we tell one normal curve from another?
• Mean tells you where it is centered
• Standard deviation tells you how thick or narrow
  the curve will be

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Normal Curves

• The 68-95-99.7 Rule.
• 68 of cases will take on a value that is plus or
  minus one standard deviation of the mean
• 95 of cases will take on a value that is plus or
  minus two standard deviations
• 99.7 of cases will take on a value that is plus
  or minus three standard deviations

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The Central Limit Theorem

• If we take repeated samples from a population,
  the sample means will be (approximately) normally
  distributed.
• The mean of the sampling distribution will
  equal the true population mean.
• The standard error (the standard deviation of
  the sampling distribution) equals

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The Central Limit Theorem

• A sampling distribution of a statistic tells us
  what values the statistic takes in repeated
  samples from the same population and how often it
  takes them.

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Confidence Intervals

• We use the statistical properties of a
  distribution of many samples to see how confident
  we are that a sample statistic is close to the
  population parameter
• We can compute a confidence interval around a
  sample mean or a proportion
• We can pick how confident we want to be
• Usually choose 95, or two standard errors

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Confidence Intervals

• The 95 confidence interval around a sample mean
  is

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Confidence Intervals

• If my sample of 100 donors finds a mean
  contribution level of 15,600 and I compute a
  confidence interval that is
• 15,600 or - 600
• I can make the statement I can say at the 95
  confidence level that the mean contribution for
  all donors is between 15,000 and 16,200.

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Confidence Intervals

• The 95 confidence interval around a sample
  proportion is
• And the 99.7 confidence interval would be

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What Determines the Margin of Error of a Poll?

• The margin of error is calculated by

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What Determines the Margin of Error of a Poll?

• In a poll of 505 likely voters, the Field Poll
  found 55 support for a constitutional
  convention.

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What Determines the Margin of Error of a Poll?

• The margin of error for this poll was plus or
  minus 4.4 percentage points.
• This means that if we took many samples using the
  Field Polls methods, 95 of the samples would
  yield a statistic within plus or minus 4.4
  percentage points of the true population
  parameter.