S ECTION 8.1 An Introduction to Confidence Intervals

S O WHAT HAS THIS ALL BEEN FOR ? Everything we have learned up to this point leads us to…. We are finally getting to the heart of Statistics! It is my favorite part!

C ONFIDENCE I NTERVALS In this chapter, we will talk about one part of inference – confidence intervals. You have seen them before and probably didn’t even know it! Ever wonder how proportions or means for the whole population are calculated if we can’t conduct a census?!?!?!?!

I T ALL STARTS WITH A … Point estimate: a statistic that provides an estimate for a population parameter. It comes from a sample. Point estimates should be CLOSE to the parameter but are seldom equal to the parameter.

L ET ’ S START BY READING SOME NEWS S TORIES Your choice!

A CTIVITY Choose one person to work with. You will be looking at the top 1000 grossing movies (as of 2011)!

V OCABULARY Confidence interval: a range for which we believe the parameter might fall. estimate ± margin of error Margin of error: how close the estimate tends to be to the unknown parameter in repeated random sampling. Confidence level (C): the overall success rate of the method for calculating the confidence interval.

I NTERPRETING C ONFIDENCE INTERVALS Fill in the blanks: We are ____% confident that the true (mean or proportion) of _____(context)_____ is between ___ and ____. Yes, you need to memorize this and be able to say it in your dreams!

E XAMPLE. Interpret the given confidence interval: The admissions director from Big City University found that (107.8, 116.2) is a 95% confidence interval for the mean IQ score of all freshmen.

E XAMPLE Find and interpret the confidence interval. A New York Times Poll asked the question, “Do you favor an amendment to the Constitution that would permit organized prayer in public schools?” Sixty-six percent of the sample answered, “Yes.” The article reported a margin of error for a 95% confidence level as 3 percentage points.

I NTERPRETING A C ONFIDENCE L EVEL ____% of all possible samples of ____(size)___ from the population will result in a confidence interval that captures the true ___(parameter)__.

E XAMPLE. Interpret the given confidence level: The admissions director from Big City University found that (107.8, 116.2) is a 95% confidence interval for the mean IQ score of all freshmen.

E XAMPLE Find and interpret the confidence level. A New York Times Poll asked the question, “Do you favor an amendment to the Constitution that would permit organized prayer in public schools?” Sixty-six percent of the sample answered, “Yes.” The article reported a margin of error for a 95% confidence level as 3 percentage points.

C HECK YOUR U NDERSTANDING p. 476

A CLOSER LOOK AT CALCULATING C ONFIDENCE INTERVALS estimate ± critical value · sample st. dev. Note: The critical value is based on the confidence level

W HAT HAPPENS TO THE MARGIN OF ERROR ?!?!?!? If the confidence level increases… So as the critical value increases? Why does this make sense? If the sample size increases? Why does this make sense?

What would you think about this confidence level?!?!?!?

H OMEWORK p. 481 (12, 16, 21, 22, 23, 24)