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practicefinal+sol

# practicefinal+sol - z-value given by 1000-1005 20 50 =-1...

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Practice Problems for the Final (only for the last topics covered in class) Problem 1. Find the likelihood estimator of the mean of a normal distribution N ( μ, 1) , knowing that sampled 10 times returned the values 10 , 8 , 9 , 11 , 12 , 9 , 7 , 8 , 11 , 10 . Solution. If the sample is x 1 , . . . , x 10 , then we need to maximize 1 2 π 10 e - P 10 i =1 ( μ - x 1 ) 2 / 2 This is done for ˆ μ = ¯ x = 9 . 5 . Problem 2. The standard deviation of the test scores on a certain test is 11.3. If a random sample of 81 students has a sample mean of 74.6, finda 90 percent confidence interval for the average score of all students. Answer. [72 . 21 , 75 . 74] . Problem 3. To find the height of a mountain, a repeated number of measurements are made. If the average of 50 independent measurements is 1000 ft, and the sample standard deviation in this case is 20 ft, can the real height be more than 1005 ft? Solution. This is a hypothesis testing and the hypothesis in this case is H 0 : μ 1005. Under this hypothesis the measured value of 1000 has the

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Unformatted text preview: z-value given by 1000-1005 20 / 50 =-1 . 7677. Therefore the P value of the test is 0 . 0385555. Therefore we accept the hypothesis at any signicance level less than . 038, meaning that the real value of the height is less than 1005. Problem 4. The height and weight of a sample of 10 people from a certain population is given below (in ft and lb): Height 5.1 5.3 4.9 5.3 5.2 5.1 4.9 5.1 4.7 5.2 Weight 105 120 97 125 108 111 98 112 98 129 Find the least-squares line for the data. Is it plausible that the weights and the heights are correlated? Solution. The least square line is y =-140 . 07 + 49 . 29 x . The correlation number is given by r = 0 . 8330452 which indicates a good linear dependence. The plot with the line in it is given bellow: 1 2...
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