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Let y denote the jth observation corresponding to the ith level of factor A and Yij the corresponding random variate.

Let y denote the jth observation corresponding to the ith level of factor A and Y

Define the linear model for the sample data obtained from the experiment by the equation

Where µ represents the general mean effect which is fixed and which represents the general condition of the experimental units, a

The last component of the model e

For the realization of the random variate Y

The expected value of the general observation y

With y

Here we may expect µ

On substitution for µ

Using (1). The normal equations can be obtained by partially differentiating E with respect to µ and

Where N = nk. We see that the number of variables (k+1) is more than the number of independent equations (k). So, by the theorem on a system of linear equations, it follows that unique solution for this system is not possible.

After carrying out some calculations and using the normal equations (2) and (3) we obtain

The first term in the RHS of equation (6) is called the

for measuring the variation due to treatment (controlled
factor), we consider the null hypothesis that all the treatment effects are
equal.

Proceeding as before, we get the residual sum of squares for this hypothetical model as

Actually,

The expression in (8) is usually called the

When actually the null hypothesis is true, if we
reject it on the basis of the estimated value in our statistical analysis, we
will be committing **Type – I Error**.
The probability for committing this error is referred to as** **the denoted by **α**. The
testing of the null hypothesis *H*_{o}
may be carried out by F test. For given **α**,
we have

i.e., It follows F distribution with degrees of
freedom K-1 and N-K.

Tags : Research Methodology - Analysis Of Variance

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