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

Section 3.1: Forecasting the Future Section 3.2: What a Sample Reveals about a Population. Prediction Interval. A prediction interval uses a population proportion to estimate an interval of sample proportions .

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

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

  2. 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.

  3. Formula for a 95% PI • So a 95% PI to estimate is: which is the same as

  4. 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 player’s next 50 times at the free-throw line.

  5. 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 we’re usually more interested in computing CI’s rather than PI’s.

  6. Confidence Intervals • CI’s 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.

  7. Confidence Intervals in the News • Consider the study on drinking habits http://poll.gallup.com/content/?ci=21307 which was conducted by the Gallup organization.

  8. 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.

  9. 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%?

  10. Finding 95% CI’s • 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.

  11. Recap from Chapter 2 • What we’ve 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.

  12. Finding Standard Error • So the only work that’s left in order to find the CI is to compute the standard error. • Recall the formula for standard error is:

  13. Problem? • What is “ ” in the previous formula? Isn’t this the quantity that we are trying to estimate? • If we don’t know the population proportion, the only reasonable estimate of it is to use the sample proportion, .

  14. Estimated Standard Error • So the formula for the estimated standard error is: • Find the estimated s.e. for the “drinking habits” example.

  15. 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:

  16. 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 can’t say it’s exactly equal to . But, as long as it’s a random sample it should be close.

  17. Margin of Error • Margin of error tells us “how close”. The margin of error for a 95% CI is

  18. 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

  19. 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.

  20. 68% Confidence Interval • If the 95% confidence interval formula is can you guess what the 68% confidence interval formula is?

  21. 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?

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