test207_sol

# test207_sol - Solution for midterm 2 1 Consider the...

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Solution for midterm 2 1. Consider the following random sample: 0.3, 0.7, 1.4, 0.9, 2.5, 3.0, 2.1, 2.4, 2.8, 0.5, 1.1, 2.6. Use the goodness-of-ﬁt test to check whether it is reasonable to believe, at the signiﬁcance level of 0.05, that this sample is from a uniform distribution on [0 ,a ] where a is an unknown parameter. Hint : use max i X i as the estimator for a . Solution: We estimate the parameter a as ˆ a = 3 . 0. Since n = 12 we can use as many as 4 bins. Since the hypothesized distribution is uniform, the bins should be the same size. That makes them [0 , 0 . 75], [0 . 75 , 1 . 5], [1 . 5 , 2 . 25] and [2 . 25 , 3 . 0], respectively. So the observed frequencies are O 1 = 3, O 2 = 3, O 3 = 1, O 4 = 5. All expected frequencies are equal to 12 / 4 = 3. So, the test statistic is χ 2 0 = 1 3 (0 + 0 + 4 + 4) = 2 . 67 , and since χ 2 0 < χ 2 0 . 05 , 4 - 1 - 1 = χ 2 0 . 05 , 2 = 5 . 99, we conclude that the null hypothesis cannot be rejected, and it is reasonable to believe that the sample came from a uniform distribution. 2. The probability theorist has constructed a simple regression model for the price of

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test207_sol - Solution for midterm 2 1 Consider the...

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