Supplementary Exercise 10.12 of IPS7e
-------------------------------------

Measurements of degree-days and gas consumption for 9 months of a year. Both
variables are response variables, but the interest is in viewing the gas
consumption as a function of the degree-days. The statistical model is a linear
regression model:
  gas_i = beta0 + beta1 * degree_i + eps_i
where the errors (eps_i) are i.i.d. from N(0,sigma).

Minitab commands for regression and fitted line plot:

MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 10\ex10_012.mtw".
Retrieving worksheet from file: 'H:\VHM\VHM801\Datasets\Minitab\Chapter
10\ex10_012.mtw'
Worksheet was saved on 07/11/2014

MTB > Fitline 'Gas Consumption' 'Degree-days';
SUBC>   Confidence 95.0;
SUBC>   Resid  'RESI1'.
Regression Analysis: Gas Consumption versus Degree-days 

The regression equation is
Gas Consumption = 1.232 + 0.2022 Degree-days

S = 0.434537   R-Sq = 97.8%   R-Sq(adj) = 97.5%

Analysis of Variance

Source      DF       SS       MS       F      P
Regression   1  58.9071  58.9071  311.97  0.000
Error        7   1.3218   0.1888
Total        8  60.2289
 
Fitted Line: Gas Consumption versus Degree-days 

MTB > Regress;
SUBC>   Response 'Gas Consumption';
SUBC>   Nodefault;
SUBC>   Continuous 'Degree-days';
SUBC>   Terms C2;
SUBC>   Constant;
SUBC>   Unstandardized;
SUBC>   Tmethod;
SUBC>   Tanova;
SUBC>   Tsummary;
SUBC>   Tcoefficients;
SUBC>   Tequation;
SUBC>   TDiagnostics 0.
Regression Analysis: Gas Consumption versus Degree-days 

Analysis of Variance

Source         DF  Adj SS   Adj MS  F-Value  P-Value
Regression      1  58.907  58.9071   311.97    0.000
  Degree-days   1  58.907  58.9071   311.97    0.000
Error           7   1.322   0.1888
Total           8  60.229

Model Summary
       S    R-sq  R-sq(adj)  R-sq(pred)
0.434537  97.81%     97.49%      96.80%


Coefficients
Term           Coef  SE Coef  T-Value  P-Value   VIF
Constant      1.232    0.286     4.31    0.004
Degree-days  0.2022   0.0114    17.66    0.000  1.00

Regression Equation
Gas Consumption = 1.232 + 0.2022 Degree-days

Fits and Diagnostics for Unusual Observations
             Gas                  Std
Obs  Consumption    Fit  Resid  Resid
  1        5.200  4.387  0.813   2.01  R

R  Large residual

Answers to questions:
---------------------
(a) gas consumption = 1.23 + 0.202 * heating days

(b) H0: beta1=0, Ha: beta1<>0
    t_obs=17.66, DF=7, P=0.0000002 (using Minitab) or P<0.0005 from the above listing
There is strong evidence against the hypothesis of no impact of number of
heating days on gas consumption. More heating days are associated with 
increased gas consumption.

(c) the 95% percentile in t(7) is 1.895, therefore the 90% CI for beta1 is
     0.2022 +- 1.895 * 0.0114 = 0.202 +- 0.022

(d) similarly, the 90% CI for beta0 is
     1.232 +- 1.895 * 0.286 = 1.23 +- 0.54