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Unformatted text preview: A Test of Normality Textbook Reference: Chapter 14.2, pages 624–26. The calculation of p-values for hypothesis testing typically is based on the assumption that the population distribution is normal. Therefore, a test of the normality assumption may be useful to inspect. A variety of tests of normality have been developed by various statisticians. One of these tests will be described here. To start, the calculation of descriptive statistics is reviewed. A data set has the numeric observations:,, . . . , . 1 x 2 x n x Familiar descriptive statistics are the sample mean: ¦ n 1i i x n 1 x and the sample variance: ¦ n 1i 2 i 2 )x x( 1 n 1 s Econ 325 – Normality Test 1 Now introduce two new statistics. The sample skewness is defined as: 2 3 2 n 1i 3 i ) ~ ( )x x( n 1 S V ¡ ¦ where ¦ V n 1i 2 i 2 )x x( n 1 ~ Skewness gives a measure of how symmetric the observations are about the mean. For a normal distribution the skewness is . A distribution skewed to the right has positive skewness and a distribution skewed to the left has negative skewness. The sample kurtosis is defined as: 22 n 1i 4 i ) ~ ( )x x( n 1 K V ¡ ¦ Kurtosis gives a measure of the thickness in the tails of a probability density function. For a normal distribution the kurtosis is 3 . Excess kurtosis is defined as: 3 K EK It follows that, for a normal distribution, the excess kurtosis is . Econ 325 – Normality Test 2 A fat-tailed or thick-tailed distribution has a value for kurtosis that...
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This note was uploaded on 01/28/2011 for the course ECON 325 taught by Professor Whistler during the Spring '10 term at UBC.
- Spring '10