Chapter 5: Analysing the Data |

## How unlikely is "unlikely"?Let us consider what a probability of 5% means. How unlikely is it? Well, imagine someone coming up to you and saying that they could select an Ace on one draw from a standard pack of cards. The chance of doing this would be = 0.067 (= about 7%). Now suppose this actually happened! At one selection from the pack, this person picked out an Ace! This seems pretty unlikely! Would you think that this was a trick (e.g., cut corners, slightly thicker cards, more than 4 Aces in the deck!) or was it just a fluke? In other words, was this likely to be a systematic event with a logical explanation or 'cause', or, "just one of those things", a random coincidence? Well in this case, we could describe our null hypothesis as being, Ho: the selection process used was random. Assuming this to be true, we then calculate the probability of drawing an Ace from 52 cards as 7%. Using statistics here as the basis for making a decision we would therefore have to conclude that this is not a "significant" event. We would retain the null hypothesis of a fair selection and conclude that the event was just a fluke (i.e., nothing can explain it other than "It was just one of those things!", or a coincidence). Even though this is a pretty unlikely event, there is not enough evidence to say that a systematic trick is going on (or that a rule has been applied). But what if this person did the same thing again!! The chance of this (assuming our null hypothesis to be true) is about 0.07 x 0.07 = 0.0049 (or about 0.5%). In this case, the probability of this event happening twice in a row (just by chance, or by Ôfair selectionÕ) is pretty remote (about one chance in 200!). We might be better advised to reject the null hypothesis and conclude that a trick of some sort is going on! That is, there is some selection rule being used here, some association or relationship is going on that will allow us to |

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