Lecture31worksheet (1)

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School
University of Notre Dame **We aren't endorsed by this school
Course
IE 310
Subject
Statistics
Date
Sep 1, 2023
Pages
9
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Lecture 31 Worksheet Chrysafis Vogiatzis Every worksheet will work as follows. 1 . You will be asked to form a group with other students in the class: you can make this as big or as small as you'd like, but groups of 4 - 5 work best. 2 . Read through the worksheet, discussing any questions with the other members of your group. You can call me at any time for help! I will also be interrupting you for general guidance and an- nouncements at random points during the class time. 3 . Answer each question (preferably in the order provided) to the best of your knowledge. 4 . While collaboration between students is highly encouraged and expected, each student has to submit their own version. 5 . You will have 24 hours (see gradescope) to submit your work. Activity 1 : Fluoride levels and cavities In the previous worksheet (from Lecture 30 ), you were asked to find the regression line for fluoride levels ( x ) and the resulting cavities ( y ). After doing all necessary calculations, we came up with the following line: ˆ y = 433.75 - 93.9 · x . 0 0.5 1 1.5 2 2.5 3 0 100 200 300 400 500 Using the line, we must have also calculated all residuals ( ˆ y i - y i ) as in the following table:
lecture 31 worksheet 2 i x i y i ˆ y i ˆ y i - y i 1 1 . 9 236 255 . 34 - 19 . 34 2 2 . 6 246 189 . 61 56 . 39 3 1 . 8 252 264 . 73 - 12 . 73 4 1 . 2 258 321 . 07 - 63 . 07 5 1 . 2 281 321 . 07 - 40 . 07 6 1 . 2 303 321 . 07 - 18 . 07 7 1 . 3 323 311 . 68 11 . 32 8 0 . 9 343 349 . 24 - 6 . 24 9 0 . 6 412 377 . 41 34 . 59 10 0 . 5 444 386 . 8 57 . 2 Problem 1 : Calculating the SS E As we noted in this lecture, the sum of squares of error SS E is an immensely useful quantity. Use the values in the last column of the table to calculate the sum of squares of error for the fluoride level regression. Answer to Problem 1 . Problem 2 : Calculating the noise variance As has become increasingly clear, the noise plays a fundamental role on how well our regression will behave. The variance of the noise can be estimated as the mean square error, which in turn is based on the sum of squares of the error. What is the noise variance in this case? Answer to Problem 2 . SSC = E,(y: -4:= (-19.34) + (56.32)+ ...: 14261.26 8 = = M), = 5 = 17x2.66
lecture 31 worksheet 3 Problem 3 : Significant regression? Combine your answer in Problem 2 (where you got the estimator for the noise variance) with your calculation of S xx = Â ( x i - x ) 2 1 to 1 Remember that x is the fluoride level. decide whether the regression is significant or not using a = 5%. Answer to Problem 3 . How about for a = 0.2%? We do not need to recalculate every- thing, right? By the way, this can prove interesting for identifying P -values, even using the T distribution critical values! x = E = 2 = 1.32 Sox = [(x x) = (0.98) - (1.285-... = 3.6h H.: B, = 0, M,: B, o To = SY 4.25 -0.025,0 2.306 Reject, Repression Strat
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