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MBA4 Regression analysis

This article discusses regression analysis in the context of a farmer's yield and hypothesis testing, including simple and multiple regression analysis. It explores the concepts of hypothesis testing, confidence intervals, and decision rules. The article provides examples and explanations to help understand the statistical analysis.

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MBA4 Regression analysis

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  1. MBA4 Regression analysis Fred Wenstøp

  2. Observation A farmer has observed his yield Y for 10 years and computes the sample mean The farmer wants to know: What is the long run average m? Theory Assume that the Ys are random numbers drawn from the same population of numbers The central limit theorem t is t-distributed around zero with n-1 degrees of freedom if the Y's are normally distributed or if the sample size is large The t-distribution Fred Wenstøp: MBA4

  3. Confidence interval Fred Wenstøp: MBA4

  4. Court trials You want to prove beyond reasonable doubt that a defendant is guilty (H1) You assume innocence (Ho) until the opposite is proven You collect evidence You pronounce the verdict If it is impossible to continue to believe in Ho in view of the evidence, Ho is rejected and H1 is pronounced to be true Otherwise, Ho is retained and pronounced to be true Statistical testing You want to establish beyond reasonable doubt (5%) that a certain hypothesis H1 is true You postulate the opposite hypothesis, Ho You collect data and compute a t You pronounce the conclusion If t turns out to be so far from zero that the probability of this is less than 5% if Ho were true, Ho is rejected and H1 is pronounced to be true Otherwise, Ho is retained Hypothesis testing, conceptual Fred Wenstøp: MBA4

  5. Hypothesis testing, operational • If m is less than or equal to 3.0, the farmer must find something else to do • Based on the observations, he wants to convince himself that m is above 3.0 He decides to use a decision rule so that the probability that he will erroneously conclude that m > 3.0 is less than a = 5% • Ho: m = mo = 3.0 • H1: m > 3.0 • He assumes Ho. • Then t is t-distributed. • He has observed t = (3.84-3.0)/0.456 = 1.84 1.83 5% 95% Fred Wenstøp: MBA4

  6. Simple regression analysis • Is it worthwhile to invest in a water sprinkling system? • He has precipitation data R for the last 10 years • Theory • Y = bo + b1R • Null hypothesis • Ho: b1 = 0 • H1: b1 > 0 b1 is estimated with an estimator b1 t = b1/sb1is t-distributed with n-2 d.f. Fred Wenstøp: MBA4

  7. Simple regression analysis • H1: b1 > 0 is not supported • Ho: b1 = 0 is retained • estimator: b1= - 0.0291 • t = b1/sb1 = -3.7761 • The analysis reveals that there actually is a significant negative correlation between rain and yield • Explanation? Fred Wenstøp: MBA4

  8. Multiple regression analysis • Explanation • Temperature also affect yield and is at the same time negatively correlated with rain! • To keep the temperature constant in the analysis of Rain, the Temp data must be included in the analysis • New theory • Y = bo + b1R + b2T • Hypotheses • Ho: b1 = 0 • H1: b1 > 0 Fred Wenstøp: MBA4

  9. Multiple regression analysis • H1: b1 > 0 is supported and Ho: b1 = 0 rejected • estimator: b1= + 0.032, t = b1/sb1 = 2.60 (t0.05 = 1.89) • For each extra mm Rain, we expect Yield to increase with 0.032 units assuming Temperature unchanged • Y = -15.84 + 0.0325Rain + 0.8805Temp Fred Wenstøp: MBA4

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