MBA4 Regression analysis Fred Wenstøp

The t-distribution Observation The farmer wants to know: Theory 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 18/01/2019 Fred Wenstøp: MBA4

Confidence interval 18/01/2019 Fred Wenstøp: MBA4

Hypothesis testing, conceptual 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 18/01/2019 Fred Wenstøp: MBA4

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% 18/01/2019 Fred Wenstøp: MBA4

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/sb1 is t-distributed with n-2 d.f. 18/01/2019 Fred Wenstøp: MBA4

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? 18/01/2019 Fred Wenstøp: MBA4

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 18/01/2019 Fred Wenstøp: MBA4

Multiple regression analysis H1: b1 > 0 is supported and Ho: b1 = 0 rejected estimator: b1= + 0.032, t = b1/sb1 = 2.60 (t 0.05 = 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 18/01/2019 Fred Wenstøp: MBA4