Georgia Institute Of Technology-Solution to Test 3 math3670-ALL ANSWERS CORRECT-GRADED A

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FALL 2018 H. Matzinger Solution to Test 3 math3670 0) a) What is the formula for the conditional probability P(AjB)? Answer: P(AjB) = P(A\ B)=P(B). b) Explain where that formula comes from. 1) We... have discovered a new method to produce a liquid containing Zinc. We measure the concentration µ four times independently (from the same solution) and find the values 0:3; 0:33; 0:3; 0:27 a) Assuming that the standard deviation of the measurement error is σ = 0:02 (when we make one measurement) estimate µ and give a 90%-confidence interval. Answer: We assume the measurement errors to be normal with zero expectation. The estimate for µ is µ^X = X1 + X2 + X3 + X4 4 = 0:3 + 0:33 + 0:3 + 0:27 4 = 0:3: The same procedure as usual leads to a confidence interval [0:3 - pbσn; 0:3 + pbσn] where σ = 0:02 and pn = p4 = 2, whilst b is defined by the equation P(-b ≤ N(0; 1) ≤ b) = 90%: The last equation above is equivalent to Φ(b) = 0:95 which we can solve with the help of a table and find b = 1:645. We have Hence the 90% confidence interval is equal to [0:3 - 0:0164; 0:3 + 0:0164] = [0:2836; 0:3164]: The true concentration is thus between 0:2836 and 0:3164 with at least 90% probability. b) Assuming the standard deviation of one measurement not known, estimate it. Answer: The sample standard deviation is our estimate and is equal to: σ^ := r02 + 0:032 + 0 3 2 + 0:032 = 0:0245 [Show More]

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