AP Statistics

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AP Statistics

• Inference Chapter 14

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Hypothesis Tests Slopes

• Given Observed slope relating Education to Job
  Prestige 2.47
• Question Can we generalize this to the
  population of all Americans?
• How likely is it that this observed slope was
  actually drawn from a population with slope 0?
• Solution Conduct a hypothesis test
• Notation slope b, population slope b
• H0 Population slope b 0
• H1 Population slope b ? 0 (two-tailed test)

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Review Slope Hypothesis Tests

• What information lets us to do a hypothesis test?
• Answer Estimates of a slope (b) have a sampling
  distribution, like any other statistic
• It is the distribution of every value of the
  slope, based on all possible samples (of size N)
• If certain assumptions are met, the sampling
  distribution approximates the t-distribution
• Thus, we can assess the probability that a given
  value of b would be observed, if b 0
• If probability is low below alpha we reject H0

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Review Slope Hypothesis Tests

• Visually If the population slope (b) is zero,
  then the sampling distribution would center at
  zero
• Since the sampling distribution is a probability
  distribution, we can identify the likely values
  of b if the population slope is zero

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Bivariate Regression Assumptions

• Assumptions for bivariate regression hypothesis
  tests
• 1. Random sample
• Ideally N gt 20
• But different rules of thumb exist. (10, 30,
  etc.)
• 2. Variables are linearly related
• i.e., the mean of Y increases linearly with X
• Check scatter plot for general linear trend
• Watch out for non-linear relationships (e.g.,
  U-shaped)

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Bivariate Regression Assumptions

• 3. Y is normally distributed for every outcome
  of X in the population
• Conditional normality
• Ex Years of Education X, Job Prestige (Y)
• Suppose we look only at a sub-sample X 12
  years of education
• Is a histogram of Job Prestige approximately
  normal?
• What about for people with X 4? X 16
• If all are roughly normal, the assumption is met

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Bivariate Regression Assumptions

• Normality

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Bivariate Regression Assumptions

• 4. The variances of prediction errors are
  identical at different values of X
• Recall Error is the deviation from the
  regression line
• Is dispersion of error consistent across values
  of X?
• Definition homoskedasticity error dispersion
  is consistent across values of X
• Opposite heteroskedasticity, errors vary with
  X
• Test Compare errors for X12 years of education
  with errors for X2, X8, etc.
• Are the errors around line similar? Or different?

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Bivariate Regression Assumptions

• Homoskedasticity Equal Error Variance

Here, things look pretty good.
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Bivariate Regression Assumptions

• Heteroskedasticity Unequal Error Variance

This looks pretty bad.
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Bivariate Regression Assumptions

• Notes/Comments
• 1. Overall, regression is robust to violations
  of assumptions
• It often gives fairly reasonable results, even
  when assumptions arent perfectly met
• 2. Variations of regression can handle
  situations where assumptions arent met
• 3. But, there are also further diagnostics to
  help ensure that results are meaningful

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Regression Hypothesis Tests

• If assumptions are met, the sampling distribution
  of the slope (b) approximates a T-distribution
• Standard deviation of the sampling distribution
  is called the standard error of the slope (sb)
• Population formula of standard error

• Where se2 is the variance of the regression error

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Regression Hypothesis Tests

• Estimating se2 lets us estimate the standard
  error

• Now we can estimate the S.E. of the slope

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Regression Hypothesis Tests

• Finally A t-value can be calculated
• It is the slope divided by the standard error

• Where sb is the sample point estimate of the
  standard error
• The t-value is based on N-2 degrees of freedom

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Regression Confidence Intervals

• You can also use the standard error of the slope
  to estimate confidence intervals

• Where tN-2 is the t-value for a two-tailed test
  given a desired a-level
• Example Observed slope 2.5, S.E. .10
• 95 t-value for 102 d.f. is approximately 2
• 95 C.I. 2.5 /- 2(.10)
• Confidence Interval 2.3 to 2.7

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Regression Hypothesis Tests

• You can also use a T-test to determine if the
  constant (a) is significantly different from zero
• But, this is typically less useful to do

• Hypotheses (a population parameter of a)
• H0 a 0, H1 a ? 0
• But, most research focuses on slopes

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Regression Outliers

• Note Even if regression assumptions are met,
  slope estimates can have problems
• Example Outliers -- cases with extreme values
  that differ greatly from the rest of your sample
• Outliers can result from
• Errors in coding or data entry
• Highly unusual cases
• Or, sometimes they reflect important real
  variation
• Even a few outliers can dramatically change
  estimates of the slope (b)

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Regression Outliers

• Outlier Example

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Regression Outliers

• Strategy for dealing with outliers
• 1. Identify them
• Look at scatterplots for extreme values
• Or, have computer software compute outlier
  diagnostic statistics
• There are several statistics to identify cases
  that are affecting the regression slope a lot
• Examples Leverage, Cooks D, DFBETA
• Computer software can even identify problematic
  cases for you but it is preferable to do it
  yourself.

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Regression Outliers

• 2. Depending on the circumstances, either
• A) Drop cases from sample and re-do regression
• Especially for coding errors, very extreme
  outliers
• Or if there is a theoretical reason to drop cases
• Example In analysis of economic activity,
  communist countries differ a lot
• B) Or, sometimes it is reasonable to leave
  outliers in the analysis
• e.g., if there are several that represent an
  important minority group in your data
• When writing papers, identify if outliers were
  excluded (and the effect that had on the
  analysis).