Lec07.Bootstrap - Bootstrap Resampling...

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Bootstrap Resampling Applied to Normal and Non-normal  Regression Data KNNL 11.5
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20 40 60 80 100 0 10 20 30 leaves Leaf Length
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The Mean (Average Leaf Length) = = N i i X N 1 1 μ = = n i i x n x 1 1 Sample Sample Population Population
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Measures of Variation for Two Different, but Related, Populations x n s s x x x n s x n i i of Variation of Variation ) ( 1 1 2 2 1 2 2 = - - = =
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) 2 / ( 1 ) 2 / ( 1 ) ( ˆ α - - ± t x E S t x t-Distribution Confidence Intervals qt(.975,30)    2.042272 qt(.025,30)   -2.042272
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To determine 95% confidence intervals from repeated sampling sample.means=rep(0,1000) for(i in seq(1000)) { sample.means[i]=            mean(sample(leaves,25)) } hist(sample.means)
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60 65 70 75 80 0 50 100 150 200 250 sample.means Distribution of Sample Means
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97.5 th quantile, 2.5 th quantile sorted.means=sort(sample.means) sorted.means[25]    63.2 sorted.means[975]    73.2 So, 95% CI is (63.2,73.2) The population mean will fall within a CI 95% of the time  under repeated sampling.
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Using single sample calculations my.sample=sample(leaves,25) mean(my.sample)      66.8 sqrt(var(my.sample)/length(my.sample))       3.109126 qt(.975,25)       2.059539 66.8-2.06*3.1    # Lower bound      60.414 66.8+2.06*3.1    # Upper bound      73.186 Compare  (63.2, 73.2)  with  (60.4, 73.2)  from above. Both are estimates of what we would expect under  repeated sampling.
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Resampling Confidence intervals, hypothesis testing  and many other types of inferences make  use of the idea of the probability under 
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Lec07.Bootstrap - Bootstrap Resampling...

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