H_{a}: The votes cast for the fifth choice are not attributable to chance alone – something else must have caused them.This is called the alternative hypothesis because it's the alternative to the status quo. In our case, as in most cases, the status quo is that nothing funny is going on and thus chance alone is the explanation for the observed behavior. The status quo is often called the null hypothesis and is denoted H_{0}:
H_{0}: The votes cast for the fifth choice are attributable to chance alone.Notice that we have defined our hypotheses to be mutually exclusive. One – and only one – must be true, and the other must be false. This is a handy property because one of the pair may be easier to accept or reject than the other. Mutual exclusivity lets us pick it and work with it, even if we're more concerned about the other. In our case, for example, we're interested in H_{a}, but it's easier to work with H_{0} because much of probability theory describes events under the control of chance alone, and that's what H_{0} proposes. So let us focus on H_{0}.
Now we can approach our problem in a more structured fashion:
- Assume that H_{0} is true.
- Ask, How unlikely is it to get eight out of twenty-two votes being cast for a single one of five choices?
- Determine this likelihood.
- Finally, if the likelihood is very small, conclude that our assumption about H_{0} is probably wrong and thus reject H_{0} (and correspondingly accept H_{a}); otherwise, accept H_{0}.
Step 1 is easy. Say it with me: We hereby assume that the events in question were caused by chance alone.
Step 2 is likewise easy. Again, say it with me: How unlikely is it to get eight out of twenty-two votes being cast for a single one of five choices?
Step 3 is, well, complicated. Twenty-two votes are cast among five, equally likely choices. One choice gets a whopping eight votes. How unlikely is this event?
To make things easier, let's consider a simpler scenario: One vote is cast among five, equally likely choices. One particular choice, the last, gets the vote. How likely is this event?
Now this question is straightforward. There are five choices, equally likely. Therefore each choice gets the vote with a 1-in-5 likelihood. If our choice in question (the last) gets the vote, we'll call the vote a "success." If it doesn't get the vote, we'll call the vote a "failure." Using these terms, the probability of a success is 1/5, and the probability of failure is 4/5.
What we have just done is created a building block for the larger analysis. A building block of this kind is often called a trial, and such trials are summarized by the single parameter p, which is simply the probability of success. (It is also customary to let q denote the probability of failure, but because this value can always be determined from p using the equation q = 1 – p, it is typically ignored until used in calculations, where notational convenience becomes important.)
At this point, let's consider a single trial. (We'll work our way up to the full twenty-two trials later.) What is the probability of a single success from a single trial? Letting n denote the number of successes and P(X) denote the probability of event X, we're asking for P(n = 1). This probability is easy to determine: It's simply p. Likewise, the probability of zero successes from one trial (in other words, a single failure) is simply q. Summarizing:
For 1 trial (N = 1):P(n = 0) = q
P(n = 1) = p
Now, let's complicate things a little by moving up to two trials. For this case, P(n = 0) is straightforward because there's only one way to get zero successes: We must fail both trials. Failing the first trial comes with probability q, and failing the second also comes with probability q. Failing both the first and second trials is therefore q
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