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test_probability_solution

# test_probability_solution - PHYS309 Spring 2010 Test on...

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PHYS309, Spring 2010 April 21, 2010 Test on Probability. Solution. Please write clearly, explain your reasoning, and submit your solution on a separate piece of paper with your name on it. Problem You are an 18-th century engineer responsible for the readiness of a cannon battery. During one of the military exercises you measure the time of flight t of the cannon balls to the target. You get 30 results (in seconds): 8.16, 8.14, 8.12, 8.16, 8.18, 8.10, 8.18, 8.18, 8.18, 8.24, 8.16, 8.14, 8.17, 8.18, 8.21, 8.12, 8.12, 8.17, 8.06, 8.10, 8.12, 8.10, 8.14, 8.09. 8.16, 8.16, 8.21, 8.14, 8.16, 8.13. You suspect that the distribution of t is Gaussian and want to check this hypothesis. To do this (a) Find the mean ¯ t and variance σ of the data. (b) Group the values of t into four bins with boundaries ¯ t - σ , ¯ t , and ¯ t + σ and count the observed number O k in each bin k = 1 , 2 , 3 , 4. Assuming the measurements were normally distributed with the mean ¯ t and variance σ calculated above, find the expected number E k in each bin. Calculate χ 2 . Is there any reason to doubt that the measurements are normally distributed?

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