With a sample of data a way to proceed is to

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With a sample of data, a way to proceed is to calculate a variance estimate as: = - - = n 1 i 2 i 2 ) x x ( 1 n 1 s Then, for the calculation of an interval estimate, replace the unknown σ with the calculated standard deviation s to get the interval estimate for the population mean as: n s x 1.645 ± A problem with this is that the confidence level is no longer guaranteed to be 0.90. The interval estimate may now be viewed as an approximate 90% interval estimate. The use of a sample variance means that the critical value 1.645 may be smaller than what will give a correct 90% confidence interval.
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Econ 325 – Chapter 7 21 It turns out that the quality of the approximation gradually improves with increasing sample size n . As a rough guideline, with n > 60, a good approximation for a 90% confidence interval is given with: n s x 1.645 ± Many economic data sets meet this requirement of a sample size exceeding 60 observations. However, methods are available for the calculation of exact interval estimates. These methods are standard features of computer software designed for the statistical analysis of economic data. Therefore, this is the next topic to discuss. Econ 325 – Chapter 7 22 The problem is to construct a confidence interval for the population mean when the population variance is also unknown. Some more statistical theory is needed. For a random sample 1 X , 2 X , . . . , n X a familiar result is: ) 1 , 0 ( N ~ n X σ μ - (the standard normal random variable) A variance estimator can be stated as: = - - = n 1 i 2 i 2 X ) X X ( 1 n 1 s Now consider the random variable: n s X t X μ - = In the denominator, σ is replaced by the estimator X s . This new random variable has a Student’s t-distribution with (n - 1) degrees of freedom . The degrees of freedom come from the divisor used for the sample variance.
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Econ 325 – Chapter 7 23 Properties of the probability density function of the t-distribution: the shape is determined by the degrees of freedom (n - 1) . like the standard normal distribution, the shape of the probability density function of the t-distribution is a symmetric curve with mean zero. But the t-distribution has thicker tails compared to the normal distribution. as the degrees of freedom increases the t-distribution becomes the same as the standard normal distribution. The graph below shows a comparison of the probability density function (PDF) of the standard normal distribution and the t-distribution with 5 degrees of freedom. -3 -2 -1 0 1 2 3 t(5) N(0,1) Note that the t-distribution is less ‘peaked’ compared to the standard normal distribution. Econ 325 – Chapter 7 24 Let ) m ( t denote a random variable having a t-distribution with m degrees of freedom. For an upper tail probability 2 α a critical value c t is the number such that: 2 ) t t ( P c ) m ( α = > Critical values are listed in an Appendix Table for the t-distribution. (Caution: when reading the tables the numeric value for the upper tail probability must be set correctly). The t-distribution critical values can also be obtained with Microsoft Excel with the function: TINV( α , degrees_of_freedom) This gives a probability of 2 α in each of the upper tail and lower tail.
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Econ 325 – Chapter 7 25 An application can proceed.
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