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Lecture2014Ch9Fa07 - Chapter 9 Estimating the Value of a...

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Chapter 9 – Estimating the Value of a Parameter Using Confidence Intervals There are two branches of statistical inference, 1) estimation of parameters and 2) testing hypotheses about the values of parameters. We will consider estimation first. Defn : A point estimate of a parameter (a numerical characteristics of a population) is a specific numerical value based on the data obtained from a sample. To obtain a point estimate of a parameter, we summarize the information contained in the sample data by using a statistic. A particular statistic used to provide a point estimate of a parameter is called an estimator . Characteristics of an Estimator To provide good estimation, an estimator should have the following characteristics: 1) The estimator should be unbiased . In other words, the expectation, or mean, of the sampling distribution of the estimator for samples of a given size should be equal to the parameter which the statistic is estimating. 2) The estimator should be consistent . In other words, if we increase the size of the sample which we select from the population, the estimator should yield a value which gets closer to the true value of the parameter being estimated. 3) The estimator should be relatively efficient . In other words, of all possible statistics that could be used to estimate a particular parameter, we want to choose the statistic whose sampling distribution has the smallest variance. It turns out that, for a population mean, μ , the sample mean, X , is an estimator which is unbiased, consistent, and relatively efficient. In any given situation, we have no way of knowing precisely how close the estimated value of a parameter is to the true value of the parameter. However, if the estimator satisfies the three properties listed above, we can be highly confident that the estimated parameter value is unlikely to differ from the true parameter value by much. On the other hand, it is nearly certain that the estimated value will not be exactly equal to the true parameter value. We want to have some idea of the precision of the estimate. Hence, rather than simply calculating a point estimate of the parameter value, we find a confidence interval estimate. Defn : A confidence interval estimate of a parameter consists of an interval of numbers obtained from a point estimate of the parameter, together with a percentage that specifies how confident we are that the true parameter value lies in the interval. This percentage is called the confidence level , or confidence coefficient . Thus, there are three quantities that must be specified: 1) the point estimate of the parameter, 2) the width of the interval (which is usually centered at the value of the point estimate), and 3) the confidence level.
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Estimation of Population Means, When σ Is Unknown
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