AP Statistics

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AP Statistics

• Chapter 11 Notes

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Significance Test Hypothesis

• Significance test a formal procedure for
  comparing observed data with a hypothesis whose
  truth we want to assess.
• Hypothesis a statement about a population
  parameter.

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Null (Ho) and Alternative (Ha) Hypotheses

• The null hypothesis is the statement being tested
  in a significance test.
• Usually a statement of no effect, no
  difference, or no change from historical values.
• The significance test is designed to assess the
  strength of evidence against the null hypothesis.
• The alternative hypothesis is the claim about the
  population that we are trying to find evidence
  for.

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Example One-sided test

• Administrators suspect that the weight of the
  high school male students is increasing. They
  take an SRS of male seniors and weigh them. A
  large study conducted years ago found that the
  average male senior weighed 163 lbs.
• What are the null and alternative hypotheses?
• Ho µ 163 lbs.
• Ha µ gt 163 lbs.

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Example Two-sided test

• How well do students like block scheduling?
  Students were given satisfaction surveys about
  the traditional and block schedules and the block
  score was subtracted from the traditional score.
• What are the null and alternative hypotheses?
• Ho µ 0
• Ha µ ? 0
• You must pick the type of test you want to do
  before you look at the data.
• Be sure to define the parameter.

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Conditions for Significance Tests

• SRS
• Normality (of the sampling distribution)
• For means
• 1. population is Normal or
• 2. Central Limit Theorem (n gt 30) or
• 3. sample data is free from outliers or strong
  skew
• For proportions
• np gt 10, n(1 - p) gt 10
• Independence (N gt 10n)

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Test Statistic

• Compares the parameter stated in Ho with the
  estimate obtained from the sample.
• Estimates that are far from the parameter give
  evidence against Ho.
• For now well us the z-test.

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P-Value

• Assuming that H0 is true, the probablility that
  the observed outcome (or a more extreme outcome)
  would occur is called the p-value of the test.
• Small p-value strong evidence against H0.
• How small does the p-value need to be?
• We compare it with a significance level (a
  level) chosen beforehand.
• Most commonly a .05

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P-value continued

• If the p-value is as small or smaller than a,
  then the data are statistically significant at
  level a.
• Ex a .05
• If the p-value is lt .05, then there is less than
  a 5 chance of obtaining this particular sample
  estimate if H0 is true.
• Therefore we reject the null hypothesis.
• If the p-value is gt .05, our result is not that
  unlikely to occur.
• Therefore we fail to reject the null hypothesis.
• If done by hand, the p-value must be doubled when
  performing a 2-sided test. The calculator will
  already display this doubled p-value if you
  choose the 2-sided option.

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Confidence vs. Significance

• Performing a level a 2-sided significance test is
  the same as performing a 1 a confidence
  interval and seeing if µ0 falls outside of the
  interval.
• e.g. If a 99 CI estimated a mean to be (4.27,
  5.12), then a significance test testing the null
  hypothesis H0 µ 4 would be significant at a
  .01.

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Reminders about Significance Tests

• 1. Dont place too much importance on
  statistically significant.
• Smaller p-value stronger evidence against H0
• 2.Statistical significance is not the same as
  practical importance.
• 3. Dont automatically use a testexamine the
  data and check the conditions.
• 4. Statistical inference is not valid for
  badly-produced data.

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Mistakes in significance testing

• Type I error
• Reject H0 when H0 is actually true.
• Type II Error
• Fail to reject H0 when H0 is actually false.

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Errors Continued
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Errors continued

• The significance level a is the probability of
  making a Type I error.
• Power The probability that a fixed level a
  significance test will reject H0 when a
  particular alternative value of the parameter is
  true.
• Ways to increase the power.
• Increase a
• Decrease s
• Increase n