slide2_1 - Parameter estimation example Suppose that out of...

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1 Parameter estimation: example Suppose that out of 1 million relays exactly  2.5% (or 25000) are defective, but we do not  know that. We would like to estimate that percentage by  inspecting a relatively small sample. What should we do?  Let’s simulate random sampling from that  populaton (can use Excel).
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2 Estimation of proportion Large population of “objects” Every “object” can be defective with  probability of  independently of the  other objects  We can select a sample of size n from  the population How can we estimate  p  from the  sample?
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3 Example 1 The 1700 ship insurer knows out of 755  ships that sailed from the local port 10  were lost at sea. What is the point  estimate    of the probability  p  that a  ship can be lost at sea? p ˆ
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4 Example 2 A company wants to estimate the  percentage of defective (out of spec)  shafts produced by a new super-fast  technology. The trial run of 500 shafts  ended up in 3 of them being out of spec.  What is the point estimate of the  proportion of defective shafts for this  process?
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5 Point estimator of proportion Sample of size  n  was taken  The number of defective items in the  sample is  X Then the point estimator of the  proportion of defective is  n X p = ˆ
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6 Why this estimator Unbiased  estimator: In other words, the long run average of  the estimated proportion is the true one p n np n X E p E = = = ) ( ) ˆ (
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7 Parameter estimation: example 2 A machine is producing shafts with the  mean diameter 0.512 mm and the  standard deviation of 0.015 mm (and  approximately normally distributed) We do not know the mean diameter and  would like to estimate What should we do?
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