## 1. Testing of hypothesis on the variance of two normal populations.

The experiment has been carried out in an attempt to disprove or reject a particular hypothesis, the null hypothesis, thus we give that one priority so it cannot be rejected unless the evidence against it is sufficiently strong.

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Now it would seem that we have to decide whether we are going to perform a hypothesis test or a significance test. To be honest, most scientists (and statisticians too!) do not always make a clear distinction between the two; probably because they usually involve largely the same calculations and tend to lead one to the same conclusion.

## In Example 4.2, the test of the null hypothesis

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The first step toward testing any of the above is to decide on the nature of the random variable to be measured and to determine its probability distribution under some defined (and relevant to our scientific hypothesis) set of conditions. For any of the above, we could look at the number of progeny obtained, although this random variable might not be the best choice in all circumstances. Note that these three scientific hypotheses make quite different assertions about the numbers of each genotype that would occur. In spite of these distinctions, the contrary, or null hypothesis, is the same, , the number of progeny is binomially distributed with a success probability of = 0.25.

## Teresa / Data-driven smooth tests when the hypothesis Is composite ..

Left unspecified in this definition, of course, is exactly what is meant by "more extreme," and whether or not its definition makes use of an alternative hypothesis or not. In our example we used a likelihood ratio criterion; but others are possible. The most obvious is simply the difference |-| where is the value of the binomial probability assigned by the null hypothesis and denotes the value(s) assigned by the alternate hypothesis. Another possibility is simply to order the possible outcomes by their probabilities under the null hypothesis as is done in the so-called "pure" test of significance (this method, of course, doesn't take into account any specific alternative hypothesis).

## Is there a simplified way to understand hypothesis ..

The power question is fairly straightforward. Clearly there is now no single power for our test of the hypothesis, but a different power for each possible value of the binomial probability included in the alternative hypothesis. The power is a function rather than a single value. This function is described in detail in the next section for Example 4.2.

## Hypothesis Testing, simple against composite

Whichever approach we are inclined to use, a complication has arisen in Example 4.2 that was not present in Example 4.1. In Example 4.2, the alternative hypothesis is composite—the alternate hypothesis is a set (actually an interval) of values of the binomial probability, not one particular value. So now how do we use the alternative hypothesis to calculate Neyman-Pearson power or to order our outcomes for a significance test?