Download Statistical Inference II: Confidence Intervals & Hypothesis Testing for Normal Populations and more Exams Introduction to Econometrics in PDF only on Docsity! Statistical Inference II: The Principles of Interval Estimation and Hypothesis Testing The Principles of Interval Estimation and Hypothesis Testing 1. Introduction In Statistical Inference I we described how to estimate the mean and variance of a population, and the properties of those estimation procedures. In Statistical Inference II we introduce two more aspects of statistical inference: confidence intervals and hypothesis tests. In contrast to a point estimate of the population mean β, like b = 17.158, a confidence interval estimate is a range of values which may contain the true population mean. A confidence interval estimate contains information not only about the location of the population mean but also about the precision with which we estimate it. A hypothesis test is a statistical procedure for using data to check the compatibility of a conjecture about a population with the information contained in a sample of data. Continuing the example from Statistical Inference I, suppose airplane designers have been basing seat designs based on the assumption that the average hip width of U.S. passengers is 16 inches. Is the information contained in the random sample of 50 hip measurements compatible with this conjecture, or not? These are the issues we consider in Statistical Inference II. 2. Interval Estimation for Mean of Normal Population When 2σ is Known Let Y be a random variable from a normal population. That is, assume ( )2~ ,Y N β σ . Assume that we have a random sample of size T from this population, 1 2, , , TY Y Y . The least squares estimator of the population mean is 1 T i i b Y T = = ∑ (2.1) This estimator has a normal distribution if the population is normal, ( )2~ ,b N Tβ σ (2.2) For the present, let us assume that the population variance 2σ is known. This assumption is not likely to be true, but making it allows us to introduce the notion of confidence intervals with few complications. In the next section we introduce methods for the case when 2σ is unknown. We can create a standard normal random variable from (2.2) by subtracting the mean and dividing by the standard deviation, ( ) 2 ~ 0,1 b b Z N TT − β − β= = σσ (2.3) Statistical Inference II: The Principles of Interval Estimation and Hypothesis Testing 2 The standard normal random variable Z has mean 0 and variance 1. That is, ( )~ 0,1Z N . Let zc be a “critical value” for the standard normal distribution, such that α = .05 of the probability is in the tails of the distribution, with α/2 = .025 of the probability in each tail. From Table 1 at the end of UE/2 the value of zc = 1.96 when α = .05. This critical value is illustrated in Figure 1. Figure 1 α = .05 critical values for the ( )0,1N distribution Thus [ ] [ ]1.96 1.96 0.025P Z P Z≥ = ≤ − = (2.4) and [ ]1.96 1.96 1 .05 .95P Z− ≤ ≤ = − = (2.5) Substitute (2.3) into (2.5) to obtain 1.96 1.96 .95 b P T − β− ≤ ≤ = σ (2.6) Multiplying through the inequality inside the brackets by Tσ yields 1.96 1.96 .95P T b T − σ ≤ − β ≤ σ = (2.7) Subtracting b from each of the terms inside the brackets gives 1.96 1.96 .95P b T b T − − σ ≤ −β ≤ − + σ = (2.8) Multiplying by −1 within the brackets reverses the direction of the inequalities giving 1.96 1.96 .95P b T b T − σ ≤ β ≤ + σ = (2.9)
