MBA 4 Regression analysis Fred Wenstp MBA 4
MBA 4 Regression analysis Fred Wenstøp: MBA 4
The t-distribution • Observation • The central limit theorem – A farmer has observed his yield Y for 10 years and computes the sample mean • The farmer wants to know: – 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 – What is the long run average m? • Theory – Assume that the Ys are random numbers drawn from the same population of numbers 3/9/2021 Fred Wenstøp: MBA 4 2
Confidence interval 3/9/2021 Fred Wenstøp: MBA 4 3
Hypothesis testing, conceptual • Court trials • Statistical testing – You want to prove beyond reasonable doubt that a defendant is guilty (H 1) – You assume innocence (Ho) until the opposite is proven – You collect evidence – You pronounce the verdict – You want to establish beyond reasonable doubt (5%) that a certain hypothesis H 1 is true – You postulate the opposite hypothesis, Ho – You collect data and compute a t – You pronounce the conclusion • If it is impossible to continue to believe in Ho in view of the evidence, Ho is rejected and H 1 is pronounced to be true • Otherwise, Ho is retained and pronounced to be true 3/9/2021 Fred Wenstøp: MBA 4 • 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 H 1 is pronounced to be true • Otherwise, Ho is retained 4
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% 1. 83 5% • Ho: m = mo = 3. 0 • H 1: m > 3. 0 95% – He assumes Ho. • Then t is t-distributed. – He has observed t = (3. 84 -3. 0)/0. 456 = 1. 84 3/9/2021 Fred Wenstøp: MBA 4 5
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 + b 1 R – Null hypothesis • Ho: b 1 = 0 • H 1: b 1 > 0 b 1 is estimated with an estimator b 1 t = b 1/sb 1 is t-distributed with n-2 d. f. 3/9/2021 Fred Wenstøp: MBA 4 6
Simple regression analysis H 1: b 1 > 0 is not supported Ho: b 1 = 0 is retained estimator: b 1= - 0. 0291 t = b 1/sb 1 = -3. 7761 The analysis reveals that there actually is a significant negative correlation between rain and yield • Explanation? • • • 3/9/2021 Fred Wenstøp: MBA 4 7
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 + b 1 R + b 2 T • Hypotheses – Ho: b 1 = 0 – H 1: b 1 > 0 3/9/2021 Fred Wenstøp: MBA 4 8
Multiple regression analysis • • H 1: b 1 > 0 is supported and Ho: b 1 = 0 rejected estimator: b 1= + 0. 032, t = b 1/sb 1 = 2. 60 (t 0. 05 = 1. 89) For each extra mm Rain, we expect Yield to increase with 0. 032 units a Y = -15. 84 + 0. 0325 Rain + 0. 8805 Temp 3/9/2021 Fred Wenstøp: MBA 4 9
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