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Unformatted text preview: Probability and Statistics II M-312 Lecture 3 3. Statistical Intervals based on a Single Sample 3.1. Basic Properties of Confidence Intervals The basic concepts and properties of confidence intervals (CIs) are more easily introduced by first focusing on a simple, albeit somewhat unrealistic, problem situation. Suppose that the parameter of interest is a population mean μ and that 1. The population distribution is normal 2. The value of the population standard deviation σ is known. Example 3.1. An article reports on a study of preferred height for an experimental keyboard with large forearm-wrist support. A sample of n = 31 trained typists was selected, and the preferred keyboard height was determined for each typist. The resulting sample average preferred height was . . 80 cm x = Assuming that the preferred height is normally distributed with σ = 2.0 cm (a value suggested by data in the article), obtain a CI for μ , the true average preferred height for the population of all experienced typists. The actual sample observations n x x , , 1 are assumed to be the result of a random sample n X X , , 1 from a normal distribution with mean value μ and standard deviation σ. The results of Probability and Statistics I imply that irrespective of the sample size n , the sample mean X is normally distributed with expected value μ and standard deviation n σ . Standardizing X by first subtracting its expected value and then dividing by its standard deviation yields the standard normal variable n X Z σ μ- = Because the area under the standard normal curve between -1.96 and 1.96 is .95, 95 . 96 . 1 96 . 1 = <- <- n X P σ μ This equation is equivalent to the following 95 . 96 . 1 96 . 1 = + < <- n X n X P σ μ σ This is a random interval having left endpoint n X σ 96 . 1- and right endpoint n X σ 96 . 1 + . In interval notation, this becomes +- n X n X σ σ 96 . 1 , 96 . 1 (3.1) The interval (3.1) is random because the two endpoints of the interval involve a random variable. It is centered at the sample mean X and extends n σ 96 . 1 to each side of X . Thus the interval’s width is n σ 96 . 1 2 ⋅ , which is not random; only the location of the interval (its midpoint X ) is random. Now (3.1) can be rephrased as Definition 3.1. If after observing , , , 1 1 n n x X x X = = we compute the observed sample mean x and then substitute x into (3.1) in place of X , the resulting fixed interval is called a 95% confidence interval for μ. This CI can be expressed as +- n x n x σ σ 96 . 1 , 96 . 1 is a 95% CI for μ or as n X n X σ μ σ 96 . 1 96 . 1 + < <- with 95% confidence....
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