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Course: MATH 3338, Fall 2008School: U. Houston
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Exam Probability First MATH 3338-10853 (Fall 2006) September 13, 2006 This exam has 2 questions, for a total of 0 points. Please answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. Name and UH-ID: 1. (a) Let x1 , . . ., xn be a sample and c be a constant. Prove that 1. if y1 = x1 + c, . . ., yn = xn + c, then s2 = s2 , and y...
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Exam
Probability First MATH 3338-10853 (Fall 2006) September 13, 2006
This exam has 2 questions, for a total of 0 points. Please answer the questions in the spaces provided on the question sheets. If you run out of room for an answer, continue on the back of the page. Name and UH-ID: 1. (a) Let x1 , . . ., xn be a sample and c be a constant. Prove that 1. if y1 = x1 + c, . . ., yn = xn + c, then s2 = s2 , and y x 2 2 2 2. if y1 = c x1 , . . ., yn = c xn , then sy = c sx , where s2 is the sample variance of the xs and s2 is the sample variance of the ys. x y (b) The article A Thin-Film Oxygen Uptake Test for the Evaluation of Automotive Crankcase Lubricants reported the following data on oxidation-induction time (min) for various commercial oils: 87 103 130 160 180 195 132 145 211 105 145 153 152 138 87 99 93 119 129 1. Calculate the sample variance and standard deviation. 2. If the observations were re-expressed in hours, what would be the resulting values of the sample variance and sample standard deviation? Answer without actually performing the re-expression. Solution: (a) 1. First, y= Then s2 y 1 = n1 1 (yi ) = y n1 i=1
2 n n
1 n
n
yi =
i=1
1 n
n
(xi + c) =
i=1
1 n
n
xi + c = x + c.
i=1
i=1
1 (xi + c ( + c)) = x n1
2 n n
n
(xi )2 = s2 . x x
i=1
2. First, 1 y= n Then s2 y 1 = n1 1 (yi y ) = n1 i=1
2 n n n i=1
1 yi = n
1 (c xi ) = c n i=1
xi = c x.
i=1
i=1
1 (c xi c x) = c n1
2 2
n
(xi )2 = c2 s2 . x x
i=1
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First Exam (continued) (b) 1. First compute
MATH 3338-10853 (Fall 2006)
September 13, 2006
xi = 87 + 103 + 130 + 160 + 180 + 195 + 132 + 145 + 211 + 105 +145 + 153 + 152 + 138 + 87 + 99 + 93 + 119 + 129 = 2563 x2 = 872 + 1032 + 1302 + 1602 + 1802 + 1952 + 1322 + 1452 + 2112 + 1052 i +1452 + 1532 + 1522 + 1382 + 872 + 992 + 932 + 1192 + 1292 = 368501 Then s2 = x and sx 35.564 2. If yi are the observations reexpressed in hours, then yi = cxi So s2 = c2 s2 1264.8/602 0.35133 y x and sy = csx 35.564/60 0.59273 with c = 1/60. 1 n1 x2 i 1 n xi
2
= (368501 25632 /19)/18 1264.8
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First Exam (continued)
MATH 3338-10853 (Fall 2006)
September 13, 2006
2. (a) Prove that for any events A and B, we have (A B) = A B (A B) = A B (these are called De Morgans laws) (b) Use Venn diagrams to verify De Morgans laws....
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