## After completing this module, the student will be able to:

### Remember, to be a valid hypothesis, the hypothesis must be refutable.

The second formulation of CH asserts that every well-ordering ofthe reals has order type less than ℵ_{2}. For a givenpointclass Γ in the hierarchy, the corresponding definablewell-ordering version of CH asserts that every well-ordering (coded bya set) in Γ has order type less than ℵ_{2}.

### pointis seldom to be disregarded.

There is no mystery about the size of 2^{} forfinite . The first natural question then is where2^{ℵ0} is located in the aleph-hierarchy: Is itℵ_{1}, ℵ_{2}, …, ℵ_{17}or something much larger?

Personally, the writer prefers to seta low standard of significance at the 5 per cent point, and ignoreentirely all results which fail to reach this level.

## The ANOVA table above is organized as follows.

are not rejecting the null when you should have. It would be like if the alarm did not go off for an armed person or telling a patient they have no disease when it turns out they do. We do not know the probability of making a type II error, and it is

## and, for any specifiable cardinal κ, let

Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.

## Is it possible to prove a research hypothesis? | ellies1mpson

impossible to calculate it. We can have a general idea about whether it is going up or down, but we can never calculate it. We do have one interesting fact, though: as the probability of type I error goes up, the probability of type II error goes down.

## ‘All swans are white cannot be proved true by any ..

There are really two kinds of approaches to newaxioms—the *local* approach and the *global*approach. On the local approach one seeks axioms that answer questionsconcerning a specifiable fragment of the universe, suchas _{ω+1} or _{ω+2}, whereCH lies. On the global approach one seeks axioms that attempt toilluminate the *entire* structure of the universe of sets. Theglobal approach is clearly much more challenging. In this entry weshall start with the local approach and toward the end we shallbriefly touch upon the global approach.

## Although the null hypothesis cannot be proven true, ..

If the jury convicts an innocent man, they have committed a . That is when you mistakenly reject the null. The likelihood of doing this is the significance level. If you are willing to have 5 percent error, that is the probability of a type I error.