What Do Samples Tell Us? In Chapter 1, we talked about ways to collect data in terms of overall design of a study. In Chapter 2, we talked about ways to choose a sample – good and bad. Now, we assume we have a good sample (one that represents the population) and a well designed study and we discuss what to do with the data. A parameter is a number that describes the ____________________. A parameter is a fixed number, but in practice we don’t know its value. A statistic is a number that describes the ________________. The value of a statistic is known when we have taken a sample, but it can change from sample to sample. We often use a statistic to estimate an unknown parameter. Example 1 Voter registration records show that 18% of all voters in Philadelphia are registered as Republicans. However, a radio talk show host in Philadelphia found that of 20 local residents who called the show recently, 60% were registered Republicans. Population? Parameter? Sample? Statistic? Why the big discrepancy between the parameter and statistic?Chapter 3 Page 2 Example 2 In a September 26, 2012 article found on pewresearch.org called, “A Record One-in-Five Households Owe Student Loan Debt”, it estimates that 19% of the nation’s households based upon polling 6,500 households owed student loan debt in 2010. (This is twice the percentage from two decades prior!) Population? Parameter? Sample? Statistic? Example 3 A lot of ball bearings has an average diameter of 2.503 cm. This is within the specifications of the purchaser. The inspector inspects 100 bearings from the lot, and they have an average diameter of 2.515 cm. The lot is rejected. Population? Parameter? Sample? Statistic? Now we’ll talk about a specific parameter and statistic -- using  to estimate p.  is the sample proportion who have the trait/opinion of interest P is the population proportion who have the trait/opinion of interestChapter 3 Page 3 Example 4 Suppose a Columbia-based health club wants to estimate the proportion of Columbia residents who enjoy running. Let p = proportion of all Columbia residents who enjoy running We decide to take an SRS of n = 100 Columbia residents.  = In our SRS of n = 100 Columbia residents, 17 said that they enjoy running. The sample proportion is Suppose now that I take another SRS of Columbia residents of size n = 100 and 22 of them said that they enjoy running. Find the sample proportion, . Notice that we have two samples from the same population and our sample proportions are different from each other. Question: Are statistics from the different samples (but drawn from the same population) going to be exactly the same? Question: Is our statistic, calculated from any one sample, going to exactly match the population parameter it is attempting to estimate? The fact that our statistics will not be the same from sample to sample is called sampling variability (since every sample is going to be a little different from each other). But, we hope that as long as we have a good sampling scheme we will be estimating the population parameter fairly well when we take our one shot at estimating it. Some terms to learn before we examine this notion of sampling variability….. _____________is consistent, repeated deviation of the statistic from the parameter in the same direction when we take many samples. _______________describes how spread out the values of the statistic are when we take many samples. Our goal is to obtain small bias and small variability – a good sampling scheme will have both!Chapter 3 Page 4 Picturing bias and variabilityChapter 3 Page 5 Picturing the Sampling Distribution of  Let’s record some important observations here: To reduce bias, use random sampling. To reduce variability of your statistic when sampling with a SRS, use a larger sample. You can make the variability as small as you want by taking a large enough sample. Large random samples almost always give an estimate that is close to the truth (population parameter).Chapter 3 Page 6 Example 5 Pewinternet.org reports on November 30, 2012 in their report, “The Best (and Worst) of Mobile Connectivity” that 85% +or- 2.4% of American adults own a cell phone. Where does plus or minus 2.4% in example 5 come from? The margin of error (MOE) is a value that quantifies the uncertainty in our estimate. When using the sample proportion to estimate the population proportion, the MOE is a measure of how close we believe the sample proportion is to the population proportion. We usually report this through a confidence interval (more on this coming soon…) When using the sample proportion, , from a SRS of size n to estimate an unknown population proportion p, the (quick formula) for MOE under 95% confidence is : We’ll refine this approximate MOE formula to something a little more precise in Chapter 21 Suppose a poll interviewed 1000 people. What is the MOE of  for 95% confidence (using the quick formula)? If the sample size is 100, what is the margin of error of  for 95% confidence (using the quick formula)? Margin of Error attempts to quantify how far off our statistic is from the parameter we’re trying to estimate. MOE tells us how far off our estimate (the statistic) is from our target (the parameter), in general. We formally quantify this “offness” by combining our statistic with MOE to form a confidence interval.Chapter 3 Page 7 Form for most confidence intervals: Approximate 95% confidence interval for p is (using the quick formula for MOE of ): A confidence statement interprets a confidence interval and has two parts: the margin of error and the level of confidence. Margin of error says how close the statistic lies to the parameter. Level of confidence says what percentage of all possible samples result in a confidence interval which contains the true parameter From example 5  = 85% with MOE 2.4% – Compute and interpret the confidence intervalChapter 3 Page 8 Example 6 http://pewsocialtrends.org/2011/05/15/is-college-worth-it/ This May 2011 Pew Research survey finds that 57% of the 2142 adult Americans polled think that “the higher education system in the United States fails to provide students good value for the money they and their families spend”. Using the quick formula for MOE, compute and interpret a 95% confidence interval for