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Unformatted text preview: Home Page Title Page Contents JJ II J I Page 1 of 45 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Chapter 6: Comparison, Causality and Prediction • In this last chapter we are going to apply the tools of inference • Look at investigations which are comparisons of two groups • Look at causal investigations • Look at predicting based on data Home Page Title Page Contents JJ II J I Page 2 of 45 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Comparisons • Two different types of comparision • We are comparing two independent groups: – Two sample problem – Typical of observational studies, but not always • We are comparing on unit with itself: – Paired or Matched problem – Typical of experimental studies, but not always Home Page Title Page Contents JJ II J I Page 3 of 45 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Comparing two populations • A model is given by Y i = μ i + R, R ∼ G (0 , σ ) . (1) • To test to see if there is a difference between the two popula- tions • We test the hypothesis μ 1 = μ 2 using the data y 11 , . . . , y 1 n 1 and y 21 , . . . , y 2 n 2 . Home Page Title Page Contents JJ II J I Page 4 of 45 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Cloud seeding ● ● ● ● ● ● ● Unseeded Seeded 0500 1500 2500 (a) Rainfall ● ● ● Unseeded Seeded 2 4 6 8 (b) Log Rainfall Home Page Title Page Contents JJ II J I Page 5 of 45 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Cloud seeding • Test the hypothesis of H : μ 1 = μ 2 • Discrepancy measure based on | ¯ y 1- ¯ y 2 | • Need sampling distribution under H to calculate p-value Home Page Title Page Contents JJ II J I Page 6 of 45 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Cloud seeding-4-2 2 4 0.0 0.1 0.2 0.3 0.4 t-density, df=50 t Density Home Page Title Page Contents JJ II J I Page 7 of 45 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit Cloud seeding • To calculate the p-value we need to calculate p- value = P ( | D | > | d | ; H ) = P ( | D | > | d | ; μ 1 = μ 2 ) • The observed value of this test statistic is d = ¯ y 1- ¯ y 2 ˆ σ q 1 n 1 + 1 n 2 = 5 . 13- 3 . 99 ˆ σ q 1 26 + 1 26 • The estimate of the common standard deviation can be shown to be ˆ σ = s ( n 1- 1)ˆ σ 2 1 + ( n 2- 1)ˆ σ 2 2 n 1- 1 + n 2- 1 = r 25 × 2 . 55 + 25 × 2 . 70 50 = 1 . 62 where ˆ σ 1 is the estimated standard deviation of the seeded clouds, and ˆ σ 2 2 of the unseeded clouds. Home Page Title Page Contents JJ II J I Page 8 of 45 Go Back Full Screen Close...
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