Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 24 Goodness-of-fit Test for Distributions Moo K. Chung [email protected] April 29, 2003 Concepts 1. Knowing the exact distribution of a sample is important for statistical analysis. We want to test if a sample x 1 , · · · , x n follows a certain probability distribution P . We perform a χ 2 goodness-of-fit test on the null hypothesis H 0 : P ( X C i ) = p i for all i = 1 , · · · , k For a continuous distribution, C i are intervals. In-class problems Example 1. Let us see if the distribution of output tuft weight (Example 14.11.) follows an exponential distribution with parameter λ . Interval 0-8 8-16 16-24 24-32 32-40 40-48 48-56 56-64 64-70 Observed # 20 8 7 1 2 1 0 1 0 Solution. The density of the exponential random variable X is given by f ( x ) = 1 λ e - x/λ , x 0. Note that R 8 0 1 λ e - x/λ . = 0 . 5. Solving this, we estimate ˆ λ = 11 . 54. Based on this, we compute the expected number of elements in each category. For instance, the expected number of element in the interval [8
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