Chapter 7.pdf - Chapter 7 Key Ideas Confidence Interval Confidence Level Point Estimate Margin of Error Critical Value Standard Error t distribution

Chapter 7.pdf - Chapter 7 Key Ideas Confidence Interval...

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Chapter 7 Key Ideas Confidence Interval, Confidence Level Point Estimate, Margin of Error, Critical Value, Standard Error t distribution, Chi-Square distribution Section 7-1: Overview All of the material in Chapters 4-6 forms a foundation of what is called inferential statistics . We already dealt with inferential statistics in Chapter 10 in a regression setting. Now, we will explore estimation. Here is an outline of the main idea of inferential statistics: 1. There is a population of interest (i.e. the group we want to know something about) 2. We draw a random sample of size n from the population. 3. We compute statistics from the sample, 4. We use these statistics to estimate similar parameters in the population. For example, suppose we want to know what percentage of Denison students own a red car. To estimate this, we take a random sample of 100 students and find out what percentage of those 100 students own red cars. Then, we say that the sample percentage should to close to the actual percentage of all Denison students who own red cars. In this example: Population of Interest: All Denison Students Sample Size: n = 100 Statistic from the Sample: The percentage of students sampled who own red cars Parameter in the Population: The percentage of all Denison students who own red cars How do we actually estimate the parameter, though? General Estimation Framework Suppose we want to estimate a parameter (e.g. population proportion, population average, etc.). The first thing to notice is that it would be impossible to exactly pinpoint the value with 100% accuracy without sampling every single member of the population, since there would always be some uncertainty. As a result, the best we can do is make a guess at the true value, and then include a margin of error based on a certain level of confidence we have in our results. The estimate and the margin of error form something called a confidence interval . A confidence interval is made of 2 different parts. 1. The point estimate is the sample statistic (this is our best guess at the true parameter value given our sample). 2. The margin of error is added and subtracted from the point estimate to make the interval. It can also be subdivided into two parts: a. A critical value from a distribution (more to come on this later) b. The standard error of the point estimate (more to come on this as well) The confidence interval (CI) has this form: CI = (Point Estimate) ± (Margin of Error) = (Point Estimate) ± (Critical Value)*(Standard Error) Of course there is no guarantee that the true population parameter will be in this interval, so we have to make some sort of statement about the chances that this will be true. The confidence level is the probability that the interval actually covers the true population parameter. Often, the confidence level is denoted (1 – α), where α is the chance that it does not cover the true parameter. For example, if α = 0.05, then the confidence level is 0.95, or 95%. Thus we would say that we are 95% confident that the interval covers the parameter.

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