//  File New-Lec2-02-04-01-04.txt. Edition 7/14/2010.


//  Lab Specs
//    Window Title
         Distribution_of_Coefficient_Estimates
//    Option (0 Normal, 1 Auto, 2 Heter, 3 X-e correlation, 4 Measurement, 5 Consistency (Is this used?)
         0
//    Constant List
         40 60 10 50
//    Coefficient List
         -2 6 2 2
//    iListVarErrors
         50 500 150 500
//    iListSamples
         3 250 1 5
//    iListSamplesAutocorrelation
         3 50 1 40
//    iListSamplesMeasurement
         3 50 1 5
//    iListXMinima
         0 10 10 0
//    iListXMaxima
         20 30 10 30
//    strListFromCoef
         -1.0 4.0 .5 1.0
//    strListToCoef
         1.0 6.0 .5 3.0
//    strListRho
         -.9 .9 .3 .0
//    strListHeter
         -2.0 2.0 1.0 .0
//    strListCoefXAndError
         -.9 .9 .3 .0
//    iListXMeasErrVar
         0 100 50 0

//  Problem Specs
//    SampleSize
//    PauseCheckbox (-1 checked, 0 cleared)
//    EstType (0 Error, 1 Constant, 2 Coefficient)
//    ErrorVar
//    ConstValue
//    CoefValue 
//    xMinValue
//    xMaxValue
//    ErrTermCheckbox (0 not visible and unchecked, -1 visible and checked, 1 visible and unchecked))
//    VarEstAndFromTo (-1 VarEst visible, 0 nothing visible, 1 From-To visible)


` 3 0 2 500 50 2 0 30 0 1
Objective: Show that a reduction in the variance of the error term's
 probability distribution reduces the variance of the coefficient
 estimate's probability distribution thereby increasing an estimate's reliability.
_
Just as before, the sample size equals 3 and the range of the x's is from 0 to 30. 
 Recall that when the sample size is 3
 and the range of the x's is from 0 to 30, the values of the x's are
 5, 15, and 25. As you did previously, calculate the sum of squared
 x deviations from their mean.
`
Two new lists now appear in the lower left of the window: a From list and a To list.
 A From value of 1.0 is specified and a To value of 3.0. The From-To Percent line
 reports the percent of repetitions in which the coefficient estimate
 falls between the From value, 1.0, and the To value, 3.0. Since the actual coefficient
 value equals 2.0, the From-To Percent line reports the percent of repetitions in which the
 the coefficient estimate lies with 1.0 of the actual value. 
_
Consequently, the magnitude of the From-To percentage reflects the reliability of an estimate.
_

` 3 0 2 500 50 2 0 30 0 1
1a. Note that the variance of the error term's probability distribution equals 500.
 Using the appropriate equation, calculate the variance of the coefficient
 estimate's probability distribution. 
_
1b. The Pause checkbox is cleared. Click Start and 
 after many, many repetitions click Stop. What is the variance of the
 numerical values for the coefficient estimates? Is this consistent
 with the variance calculated from the equation (Question 1a)?
_
1c. In what percent of the repetitions does the estimate fall within 1.0 and 3.0 range?


` 3 0 2 50 50 2 0 30 0 1
2a. The variance of the error term's probability distribution has been
 reduced from 500 to 50. Using the appropriate equations, calculate the
 variance of the coefficient estimate's probability distribution.
_
2b. Click Start and after many, many repetitions click Stop. What is the variance of the
 numerical values for the coefficient estimates? Is this consistent
 with the variance calculated from the equation (Question 2a)?
_
2c. In what percent of the repetitions does the estimate fall within 1.0 and 3.0 range?

`
3. How does a decrease in the variance of the error term's probability distribution
 affect the variance of the coefficient estimate's probability
 distribution? How does a decrease in the variance of the error term's
 probability distribution affect the reliability of a coefficient estimate?
 Why does this make sense intuitively?