Question

In developing empirically a cost function from observed data on a complex chemical experiment, an analyst employed normal error regression model (Y_i = _$\beta$0 +_$\beta$1Xi+ $\varepsilon$_i ). $\beta$0 was interpreted here as the cost of setting up the experiment. The analyst hypothesized that this cost should be $7.5 thousand and wished to test the hypothesis by means of a general linear test.

a) Indicate the alternative conclusions for the test

b) Specify the full and reduced models

c) Can you tell what the quantity df_R - df_F in test statistic $F^{*} = \frac{SSE(R)-SSE(F)}{df_{R}-df_{F}} \div \frac{SSE(F)}{df_{F}}$   will equal in the analyst's test? Explain.

Answer 1

(a) The alternative conclusion for the test is the cost is should not be $7.5 thousand.

H_a: _$\beta$1 not equal to $7.5 thousand

(b) The full model or unrestricted model is the model thought to be most appropriate for the data. For simple linear regression the full model is:

Y_i = _$\beta$0 +_$\beta$1Xi+ $\varepsilon$_i , where  Y_i = is dependent variable  _$\beta$0 and _$\beta$1  are parameter estimates Xi is independent variable and

$\varepsilon$_i are errors in estimation

• The reduced or restricted model is the model described by the null hypothesis H_o.

For simple linear regression , a common null hypothesis is H0:_$\beta$1 =0. or  H₀: Y_i = _$\beta$0 + $\varepsilon$_i.

In this case, the reduced model is taken when the null hypothesis is true.

The reduced model is given by:  Y_i = _$\beta$0 + $\varepsilon$_i. where  _$\beta$0 is intercept of the model.

The reduced model suggests that the dependent or response variable Y_i is a function of sum of overall mean , _$\beta$0 and error term $\varepsilon$_i  .

(c) Suppose we have 'n' observation for dependent variable X.

Then df_R is degree of freedom of reduced model (i.e. When the null hypothesis H_o:_$\beta$1 =0 is true )

and df_F is degree of freedom of full model (i.e. When the null hypothesis H_o:_$\beta$1 =0 is false )

Here,  df_R= (n-1), since in reduced model we are estimating on one parameter _$\beta$0. So degree of freedom = n- no of estimates = n-1 in this case.

df_F= (n-2), since in full model we are estimating on two parameters _$\beta$0 and _$\beta$1. So degree of freedom = n- no of estimates = n-2 in this case.

Hence, the quantity df_R - df_F =(n-1)-(n-2)= 1