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slide3 - Qualityofpointestimation

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1 Quality of point estimation Example: a group of high school students needs to estimate the  average height of trees in the nearby  forest. They randomly  selected 10 trees, measured their heights and obtained the  following numbers: 9.5, 11.5, 8.3, 10.0, 10.2, 9.2, 9.8, 10.8, 9.9,  8.9. So the unbiased point estimate of the average height is                 but how far from this estimate is the true average height? Although (as always in statistics) we can’t answer this question  with 100% certainty, we can give an interval (the  confidence  interval ) containing the true average with high probability, say,  95%. 8 . 9 = x
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2 Quality of point estimation To achieve that we need to make use of the  probability distribution of our estimator (the  sampling  distribution ). Due to CLT, we know that       has a normal  distribution with mean       (the true average they are  trying to estimate) and standard deviation    (we will  assume       to be known to be equal to 1m).  This implies that                  is standard normal. Therefore it takes values in the interval      with probability 95%. n X Z / σ μ - = n ] , [ 2 / 05 . 0 2 / 05 . 0 z z - X
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3 Quality of point estimation Mathematically speaking, Or, writing it for the sample mean, That’s great, but we do not know      and the value of       is already known (so we do not need a probability  for it). 95 . 0 ) 96 . 1 96 . 1 Pr( = < < - Z μ x 95 . 0 96 . 1 96 . 1 Pr = + < < - n X n σ
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4 Quality of point estimation This is not a big deal though as we can rewrite the  event                                               as follows:  n X n σ μ 96 . 1 96 . 1 + < < - n X n 96 . 1 96 . 1 < - < - X n X n - < - < - - 96 . 1 96 . 1 n X n X 96 . 1 96 . 1 + < < -
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5 Quality of point estimation So we obtain the desired 95% confidence interval for  the true mean:  That is (plugging the numbers in), the true mean  height of the trees in the forest lies between         and    with probability 95%. 95 . 0 96 . 1 96 . 1 Pr = + < < - n X n X σ μ 2 . 9 6 . 0 8 . 9 10 1 96 . 1 8 . 9 = - = - 4 . 10 6 . 0 8 . 9 10 1 96 . 1 8 . 9 = + = +
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6 Quality of point estimation Let us now summarize the procedure for obtaining  confidence intervals. Establish the sampling distribution for the estimator used to 
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