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Unformatted text preview: l is given with: x ± 1.645 s n 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. 21 Chapter 8 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 X 1 , X 2 , . . . , X n a familiar result is: X−μ
~ N(0 , 1) σ
n (the standard normal random variable) A variance estimator can be stated as: s2
∑ (X i − X )2 n − 1 i=1 Now consider the random variable: t= X−μ sX
n In the denominator, σ is replaced by the estimator s X . 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. 22 Chapter 8 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. N(0,1) t(5) -3 -2 -1 0 1 2 3 Note that the t‐distribution is less ‘peaked’ compared to the standard normal distribution. 23 Chapter 8 Let t...
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- Spring '10