The Little Blue Reasoning Book - Brandon Royal [23]
Consider the question of whether daughters share the same political beliefs as their mothers. Let’s assume that exactly 100 females are surveyed. This cross-tabulation below, which is fictitious though not implausible, suggests that the second generation of females follows the basic political beliefs of the first generation.
Political Beliefs
Stockbroker Endorsement
“My broker helped me achieve an above-average return on my stock investment portfolio. His predictions turned out to be correct, whether judging the stock market index or the performance of individual companies. My friend, a seasoned businessperson, tried to predict the market himself and consistently achieved a negative return. The advice is clear. Keep your hand out of the cookie jar and don’t try to predict the stock market yourself. Use a broker and get the returns you deserve.”
How do you go about evaluating the more general claim that brokers do in fact make better stock market investment decisions than do “regular” businesspersons?
In testing this hypothesis, we employ a method based on experimental design, which utilizes a matrix consisting of two primary rows and two primary columns, with nine boxes of numerical data.
Note that in this example, percentage calculations are required because the actual numbers of predictions are of unequal size (predictions by stockbrokers total 200, while predictions by regular businesspersons total 800). The percentage of correct predictions is calculated as follows: stockbrokers: 50⁄200 = 25%; regular businesspersons: 100⁄800 = 12.5%.
The numbers in the chart above are hypothetical. However, based on these numbers, we find that brokers are twice as likely to make correct predictions (25% vs. 12.5%), and we can conclude that there is merit in the ability of brokers, as compared with regular businesspersons, to make accurate stock market predictions. It is especially important to think not just in terms of the number of correct predictions made, but of the percentage of correct predictions made over both categories (i.e., the percentage of correct predictions made by stockbrokers versus the percentage of correct predictions made by regular businesspersons).
For a comparative problem, refer to Shark.
Hypothesis testing is about making predictions. By the word “hypothesis” we mean “a statement yet to be proven.” For example, let us say we are on our way to the doctor’s office for a major checkup. In particular, we are concerned about the possibility that we might have cancer, but obviously, we know that this is a bit unlikely. So we enter our checkup with the hypothesis: “I do not have cancer.”
Upon completion of tests, we will be diagnosed either as having cancer or not. In reality we may or may not have cancer, and the tests may or may not confirm this. This creates four possibilities. The hypothesis to be tested may be true or false, and we may accept or reject it. In other words, we may accept a hypothesis that is true or false, or reject a hypothesis that is true or false. The possibilities may be shown in diagram form:
Generic Outline for Hypothesis Testing
With respect to the chart above: TA stands for “acceptance of a true hypothesis,” TR stand for “rejection of a true hypothesis,” FA stands for “acceptance of a false hypothesis,” and FR stands for “rejection of a false hypothesis.” Naturally, we wish to avoid the rejection of a true hypothesis, known as a Type I error, as well as avoid the acceptance of a false hypothesis, known as a Type II error.
Hypothesis testing will always involve the possibility of Type I and Type II errors. The risk of one of these errors will always be deemed greater than the other. Let’s look at the hypothesis: “I do not have cancer.” In this case, suppose the hypothesis is true and we reject it. We have committed a Type I error. Now suppose the hypothesis is incorrect and we accept it. Then we have committed a Type II error. Here, the Type II error is more serious