1. MBA4 Regression analysis
Fred Wenstøp

2. 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

3. Confidence interval
18/01/2019
Fred Wenstøp: MBA4

4. 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

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 = ( )/0.456 = 1.84
1.83 5%
95%
18/01/2019
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/sb1 is t-distributed with n-2 d.f.
18/01/2019
Fred Wenstøp: MBA4

7. Simple regression analysis
H1: b1 > 0 is not supported
Ho: b1 = 0 is retained
estimator: b1=
t = b1/sb1 =
The analysis reveals that there actually is a significant negative correlation between rain and yield
Explanation?
18/01/2019
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
18/01/2019
Fred Wenstøp: MBA4

9. Multiple regression analysis
H1: b1 > 0 is supported and Ho: b1 = 0 rejected
estimator: b1= , t = b1/sb1 = 2.60 (t 0.05 = 1.89)
For each extra mm Rain, we expect Yield to increase with units assuming Temperature unchanged
Y = Rain Temp
18/01/2019
Fred Wenstøp: MBA4