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.

• A 95% (68%) PI for a sample proportion is from 2 (1) standard error below the population proportion to 2 (1) standard error above.

Formula for a 95% PI• So a 95% PI to estimate is:

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

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.

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.

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.

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.

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%?

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.

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.

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:

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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, .

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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 can’t say it’s exactly equal to . But, as long as it’s 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.

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?