How did the assumption of independence simplify the

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How did the assumption of independence simplify the variance formula ? That is, n ) X ( Var 2 σ = This gives the result that as the sample size n increases the variance of the sample mean decreases. The standard deviation of the sampling distribution of X is called the standard error of X . This is: n ) X ( se σ =
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Econ 325 – Chapter 6 5 Now introduce the assumption of normality. Let the random sample 1 X , 2 X , . . . , n X be a set of normally distributed and independent random variables with mean μ and variance 2 σ . It follows that X is also normally distributed (recall that an earlier result stated that a linear combination of normally distributed random variables is also normally distributed). That is, σ μ n , N ~ X 2 PDF of X for n=25 and n=100 n = 100 n = 25 Note: the total area under each curve is equal to one. Econ 325 – Chapter 6 6 The graph on the previous page shows the probability density function of the sampling distribution of the sample mean. This is centered at μ . The graph demonstrates that as the sample size n increases, the variance decreases, and the distribution becomes more concentrated around the population mean. A standardized normal random variable can be stated: ) 1 , 0 ( N ~ n X ) X ( se X Z σ μ - = μ - = Probability statements about the mean can now be considered.
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Econ 325 – Chapter 6 7 Example Times spent studying by students in the week before final exams follow a normal distribution with standard deviation 8 hours. A random sample of 4 students was taken in order to estimate the mean study time for the population of all students. Questions and Answers (a) What is the probability that the sample mean exceeds the population mean by more than 2 hours ? That is, find: ) X ( P 2 + μ > With n = 4 the standard error of the sample mean X is: 4 2 8 = = σ = n ) X ( se Write the problem as a probability statement about the standard normal random variable Z : ( ) ) 5 . 0 ( F 1 ) 5 . 0 Z ( P 1 Z P ) X ( se ) ( ) X ( se X P ) X ( P - = < - = > = μ - 2 + μ > μ - = 2 + μ > symmetry by 4 2 A look-up in the Standard Normal Distribution Table gives: 0.6915 = ) 5 . 0 ( F The answer is: 0.3085 0.6915 = - = 2 + μ > 1 ) X ( P Econ 325 – Chapter 6 8 (b) What is the probability that the sample mean is more than 3 hours below the population mean ?
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