Chapter 20: Testing Hypotheses about Proportions

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Chapter 20Testing Hypotheses about Proportions

• Half the money I spend on advertising is wasted
  the trouble is I dont know which half.
• -John Wanamaker

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Hypotheses

• Hypothesis
• a supposition
• a proposition or principle which is supposed or
  taken for granted, in order to draw a conclusion
  or inference for proof of the point in question
• In statistical tests of hypotheses, we assume
  that a hypothesis is true.
• If the data are consistent with the hypothesis,
  we retain the hypothesis
• If the data are not consistent with the
  hypothesis, we reject the hypothesis

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Testing Hypotheses

• The Null Hypothesis
• The starting hypothesis
• The null hypothesis specifies a population
  model parameter of interest and proposes a value
  for the parameter
• Standard Deviation (not Standard Error)
• We assume the null hypothesis to be true, so we
  have a value for the model parameter p.

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The Reasoning of Hypothesis Testing

• Hypotheses
• The null hypothesis
• Translate our question into a statement about
  model parameters
• Write
• The alternative hypothesis
• Contains the values of the parameter we accept if
  we reject the null.
• Write

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The Reasoning of Hypothesis Testing

• Plan
• Specify the model you will use to test the null
  hypothesis and the parameter of interest
• All models require assumptions, so state them and
  check any corresponding conditions
• Include the name of the test you plan to perform
• End with a statement such as
• Because the conditions are satisfied, it id
  appropriate
• to model the sampling distribution of the
  proportion with a
• model

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The Reasoning of Hypothesis Testing

• Mechanics
• Do the actual calculations of a test statistic
  from the data
• The ultimate goal is to obtain a P-value
• The probability that the observed statistic value
  could occur if the null model were correct
• If the P-value is small enough, reject the null
  hypothesis

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The Reasoning of Hypothesis Testing

• Conclusion
• The conclusion in a hypothesis test is always a
  statement about the null hypothesis
• reject or fail to reject the null hypothesis
• Consider the size of the effect by examining a
  confidence interval

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One-proportion z-test

• The conditions for the one-proportion z-test are
  the same as for the one-proportion z-interval.

When the conditions are met and the null
hypothesis is true, the statistic follows the
standard Normal model, so we can use that model
to obtain a P-value.
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Alternative Alternatives

• Two-sided alternative
• When we are equally interested in proportions
  that deviate from p in either direction

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Alternative Alternatives

• One-sided alternative
• An alternative hypothesis that focuses on
  deviations from the null hypothesis value in only
  one direction.
• The P-value is the probability of deviating only
  in the direction of the alternative away from the
  null hypothesis value

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TI-83 Tips

• STAT TESTS
• 5 1-Prop ZTest

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TI-83 Tips

• Specify the hypothesized proportion Po
• Enter x, the observed number of wins
• Specify the sample size
• Decide on one- or two-tailed test
• Calculate

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P-Values and DecisionsWhat to Tell About a
Hypothesis Test

• Hypothesis tests are useful when making a
  decision
• It is a good idea to report the confidence
  interval for the parameter of interest
• The P-value is highly context-dependent
• The importance of the issue also factors into the
  choice of P-value
• The conclusion about any null hypothesis should
  be accompanied by the P-value of the test. If
  possible, it should also include a confidence
  interval for the parameter of interest