Quantitative Methods by Brandimarte Paolo;

Author:Brandimarte, Paolo;
Language: eng
Format: epub
ISBN: 698852
Publisher: Wiley
Published: 2011-12-25T16:00:00+00:00

against the alternative Ha that not all expected values are the same. Another key assumption concerns population variances. They are unknown, but it is assumed that all of them have the same value σ. This might seem a bold assumption, but keep in mind that we want to check the equality of the expected values or, more informally, if there is any significant difference among the populations; hence, in terms of null hypothesis, it is natural to assume the same variance.

Since we have m samples of size n, we have a grand total of nm independent, normally distributed observations. If we standardize, square, and add all of them, we obtain a chi-square random variable with nm degrees of freedom

(9.33)

Since expected values are unknown, we should replace them with sample means for each population

(9.34)

where the notation i, points out that this is a sample mean obtained by summing over the second subscript j. If we plug these sample means into Eq. (9.33), we get the random variable

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