Chapters6_7.pdf - Chapter 6 Confidence Intervals This...

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100 Chapter 6 Confidence Intervals This chapter introduces a technique for statistical inference, confidence intervals . 6.1 Introduction 1. As stated in Chapter 1, statistical inference is when we take the information from a sample and use various statistical techniques to produce conclusions about a population . We will focus on two types of statistical inference: 1. Confidence Intervals (Chapter 6) 2. Hypothesis Tests (Chapter 7) 2. Example In a Gallup poll 1 taken June 8-10, 2015 of 1500 people, it was found t hat 43% of the sample approved of President Obama’s job performance. A confidence interval allows us to make a conclusion about the population of voters: The confidence interval above is interpreted to mean that in the entire population, the true proportion of voters who approve of the president’s job performance is between 40% and 46% at the 95% confidence level (we explain confidence level shortly). Notice that other (CNN, Fox, etc. ) polling agencies 2 obtained results that are mostly in this range over the same time period and same question. The sample proportions in those polls ranged from 42% to 47%. 3. Example From a student project. A 90% confidence interval on 𝜇 , the true, unknown average amount of money men spend at Walmart in Valdosta is: How do we interpret this interval? We have 90% confidence (again, we explain this shortly) that on average men spend between $20.65 and $32.21 at Walmart in Valdosta. 4. A confidence interval ( CI ) is an interval estimate about a population parameter . In this chapter we study the population parameters: 1. 𝜇 is the true, unknown mean of a population 2. ? is the true, unknown proportion of a population that have a characteristic of interest. 5. All confidence intervals have this form: ????????? ± ?????? ?? ????? When we are drawing inference on 𝜇 , is the estimator of 𝜇 . In other words, we estimate 𝜇 with the sample average. We will talk about the margin of error later. 1 http://www.gallup.com/poll/113980/Gallup-Daily-Obama-Job-Approval.aspx 2 http://www.realclearpolitics.com/polls/
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101 6. Confidence intervals include two levels of uncertainty or error: a. Margin of error sometimes called the precision of the estimate. A confidence interval is more precise when the margin of error is small and in that case we say that we have a tighter estimate of 𝜇 . b. Level of confidence We you make a confidence interval, you must choose a confidence level . Typically one of these values is used: 80%, 90%, 95%, 99%. A natural question is why not 100%? We will address that later. Let’s explain confidence level by way of an example. Suppose you have a population that you know has 𝜇 = 10 and you take a sample of size 30 from that population. Next, you decide to us a confidence level of 90% and calculate a 90% confidence: [9,13]. We note that this CI does contain the true value of 𝜇 , i.e.
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