11 Bootstrap

11 Bootstrap - Variation in the Mean Confidence Intervals...

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Unformatted text preview: Variation in the Mean Confidence Intervals Bootstrap 20 40 60 80 100 10 20 30 leaves Leaf Length 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 ( ) ∑ = − = N i i x X X N 1 2 2 1 σ ( ) ∑ = − − = n i i x x x n s 1 2 2 1 1 Population Population Sample Sample Variance ( ) ( ) ( ) ( ) 2 2 1 2 2 1 2 1 2 1 2 2 1 1 1 1 ) ( ) ( 1 1 1 1 x n i i n i i n i i n i i y s a x x n a x a ax n b x a b ax n y y n s = − − = − − = + − + − = − − = ∑ ∑ ∑ ∑ = = = = Variance of y=ax+b ) var( ) var( ) var( 2 1 2 1 x x y x x y + = + = If independent... which is true for random samples. Variance of the sum is the sum of the variances: Measures of variation for two different, but related, populations x x x n s x n s s n i i x of Variation ) ( 1 1 of Variation 1 2 2 2 2 ∑ = − − = = Variance of Mean n s x V n x n V x V s n i n i x 2 1 2 1 2 ) ( 1 1 ) ( = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = ∑ ∑ = = Hypothesis Testing ¡ Is this population different from some other population or some hypothetical population? ¡ One way to check is by comparing means. ¡ If the means are measurably different then the populations must be different. ¡ How much different do they need to be? How much different? ¡ Rule of thumb: ¢ Different if outside of what might be expected by chance. ¢ Outside of what might occur 95% of the time is perhaps too far ¢ 95% Confidence Intervals ) 2 / ( 1 ) 2 / ( 1 ) ( ˆ α α − − ± z x E S z x Normal Distribution Confidence Intervals qnorm(.975) 1.959964 qnorm(.025)-1.959964 ) 2 / ( 1 ) 2 / ( 1 ) ( ˆ α α − − ± t x E S t x t-Distribution Confidence Intervals qt(.975,30) 2.042272 qt(.025,30)-2.042272 x Probability-3-2-1 1 2 3 0.0 0.1 0.2 0.3 0.4 Distribution of Mean 95% CI x Probability-3-2-1 1 2 3 0.0 0.1 0.2 0.3 0.4 Distribution of Mean Display Types my.sample 1.0 1.5 2.0 2.5 3.0 40 50 60 70 80 Leaves Sample Population Mean +/- CI 1.0 1.5 2.0 2.5 3.0 40 50 60 70 80 Comparing Population Means 2SE Population Mean +/- CI 1.0 1.5 2.0 2.5 3.0 40 50 60 70 80 Comparing Population Means 2SE To determine 95% confidence...
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This note was uploaded on 11/12/2009 for the course BTRY 3010 at Cornell.

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11 Bootstrap - Variation in the Mean Confidence Intervals...

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