Section 3.1: Forecasting the Future Section 3.2: What a Sample Reveals about a Population

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Section 3.1 Forecasting the Future Section
3.2 What a Sample Reveals about a Population
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Prediction Interval

• A prediction interval uses a population
  proportion to estimate an interval of sample
  proportions.
• A 95 (68) PI for a sample proportion is from 2
  (1) standard error below the population
  proportion to 2 (1) standard error above.

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Formula for a 95 PI

• So a 95 PI to estimate is
  
• which is the same as

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Prediction Interval Example

• Suppose that a high school basketball player has
  a free throw shooting percentage of .80.
• Find and interpret a 95 prediction interval
  for this players next 50 times at the free-throw
  line.

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Confidence Intervals

• A confidence interval differs from a prediction
  interval in that with a CI one uses a sample
  proportion to predict an interval of values
  containing the population proportion.
• In practice were usually more interested in
  computing CIs rather than PIs.

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Confidence Intervals

• CIs for a population proportion allows you to
  estimate population proportions for a large
  population without interviewing every single
  person in the population.
• Ex Estimate the proportion of all American
  households who own at least 2 cars.

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Confidence Intervals in the News

• Consider the study on drinking habits
  http//poll.gallup.com/content/?ci21307 which
  was conducted by the Gallup organization.

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Making Sense of a Real-life CI

• Our goal is to understand the confidence
  interval language
• For results based on the total sample of national
  adults, one can say with 95 confidence that the
  maximum margin of sampling error is 3 percentage
  points.

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Finding a 95 CI

• Based on the recent survey, 29 of Americans (in
  the sample) said they only drink on special
  occasions.
• What is the appropriate symbol for 29?

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Finding 95 CIs

• 29 is a sample proportion (based on 1011
  American national adults) who responded that they
  only drink on special occasions.
• Use this statistic to find a 95 CI to estimate
  the proportion of ALL American national adults
  who only drink on special occasions.

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Recap from Chapter 2

• What weve seen so far is that whatever the
  proportion in the population, we are 95
  confident that the sample proportions fall within
  2 s.e.s of the population proportion.
• Since distances work both ways, if the sample
  proportion is within 2 s.e.s of the population
  proportion then the population proportion is
  within 2 s.e.s of the sample proportion.

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Finding Standard Error

• So the only work thats left in order to find the
  CI is to compute the standard error.
• Recall the formula for standard error is

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Problem?

• What is in the previous formula? Isnt
  this the quantity that we are trying to estimate?
  
• If we dont know the population proportion,
  the only reasonable estimate of it is to use the
  sample proportion, .

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Estimated Standard Error

• So the formula for the estimated standard error
  is
• Find the estimated s.e. for the drinking habits
  example.

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Putting it all together

• Again, since distances work both ways, if the
  sample proportion is within 2 s.e.s of the
  population proportion then the population
  proportion is within 2 s.e.s of the sample
  proportion.
• Therefore a formula for a 95 CI is

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Understanding this formula

• If you want to estimate an unknown population
  proportion, , the best way to get an
  estimate is using a sample proportion .
• Since the estimate for was only based on one
  sample we cant say its exactly equal to .
  But, as long as its a random sample it should
  be close.

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Margin of Error

• Margin of error tells us how close. The margin
  of error for a 95 CI is

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Understanding this Formula

• This formula follows from the fact that provided
  is within 2 s.e.s of , will be
  within 2 s.e.s of .
• In other words, with 95 confidence is
  located within the interval

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Back to the drinking example

• Find and interpret a 95 CI for the proportion of
  ALL American national adults who drink only on
  special occasions.

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68 Confidence Interval

• If the 95 confidence interval formula is
• can you guess what the 68 confidence interval
  formula is?

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How does the confidence level of CI affect the
interval?

• Compare and contrast the 68 and 95 confidence
  intervals.
• As the level of confidence increases/decreases,
  the width of the CI increases/decreases.
• Does this make sense?