Preparing for the next class

In the next in-class activity, you will need to understand the basic idea of null and alternative hypotheses, that an event with low probability is very unlikely but still may occur, and how probability can be used as statistical evidence.

Suppose that you are playing a game with your friend that involves flipping a coin. Each round consists of flipping the coin 10 times. Your friend is favored if more of the flips land on heads, and you are favored if more of the flips land on tails.

In this situation, you had a baseline assumption when you started playing the game with your friend: the coin is fair. This assumption is called the null hypothesis. Then you suspected that your friend’s coin was weighted in favor of heads. This guess is called the alternative hypothesis. If you wanted to claim that your friend was indeed cheating, you would need to produce evidence in order to prove that their coin was not fair.

Statistics is a very useful tool in this scenario and can be used to test these kinds of hypotheses. In a statistical hypothesis test:

• The null hypothesis, [latex]H_{0}[/latex], is what we assume to be true to begin with. It is often a statement of no change from the previous value or from what is expected (e.g., we expect a coin to be fair).
  • The null hypothesis, [latex]H_{0}[/latex], is always given in the form: [latex]parameter = value[/latex].
• The alternative hypothesis, [latex]H_{A}[/latex], is what we consider to be plausible if the null hypothesis is false. Often, it is a change from the null hypothesis that we would like to test the accuracy of. The alternative hypothesis answers the question, “Do we think the actual parameter is larger than, smaller than, or just different from the null value, where the null value is the value specified in the null hypothesis?”
  • The alternative hypothesis, [latex]H_{A}[/latex], is always given as an inequality: [latex]parameter>null value, parameter
• The evidence used is probability. The statistical evidence that we gather is always evidence in support of the alternative hypothesis and against the null hypothesis. We ask ourselves the question, “Do we have enough evidence to reject the null hypothesis?”
• The outcomes of the hypothesis test are either:
  • We reject the null hypothesis (we have gathered enough evidence).
  • We fail to reject the null hypothesis (we have not gathered sufficient evidence, so we cannot reject the starting assumption).

Your friend claims the coin is fair, but you aren’t convinced. You can’t prove for certain  that your friend’s coin is weighted unfairly (without special equipment, of course), but  you can test your hypotheses with a sample of coin flips. If the proportion of heads in  your sample is high enough, it provides strong evidence that your friend’s coin is  weighted in their favor. In other words, a high enough proportion of heads would be  sufficient evidence for you to reject the assumption that your friend’s coin is fair. Let’s  suppose that you flip the coin 20 times for your sample.

Looking ahead

Notice that above, you assumed your friend’s coin is fair, and that assumption was used  to calculate the probability of obtaining 17 or more heads. The probability of 0.0013 is  the probability of obtaining 17 or more heads out of 20 flips if the coin is fair, and this  low probability is evidence that the coin is not fair. Note that no matter what probability  you obtain, even if it is a high probability, it is not evidence that the coin is fair. This is  because fairness was already the given assumption, and the probability was computed  using the assumption that the coin is fair. If we assume something is true, we can’t use  that assumption to prove that it’s true (that would be using circular logic). From our  sample of coin flips, we either get enough evidence to say the coin is weighted, or we  don’t. In hypothesis-testing language, we either get enough evidence to reject the null  hypothesis, or we don’t. When we don’t get enough evidence to reject the null  hypothesis, we fail to reject the null hypothesis. We never accept the null hypothesis  because that was already our assumption to begin with.