Episode #4 of the course Introduction to statistics by Polina Durneva
Today, we will discuss the main ideas behind hypothesis tests and their interpretation with respect to statistical analysis.
Two Types of Hypotheses
In the previous lessons, we used confidence intervals to estimate a population parameter. For instance, if we want to know what percentage of high school seniors got accepted to college this year, we will select a random sample of high school seniors and survey them. Assuming a margin of error, we will use our estimated value and create a confidence interval, which contains the real percentage of high school seniors accepted to college. If a margin of error is 5% and 40% of our random sample was admitted to college, we can assume that between 35% and 45% of all high school seniors already got accepted to college this year.
However, we can also conduct hypothesis testing if we want to evaluate a certain claim about population. For instance, if someone were to claim that more than 50% of all high school seniors already got accepted to college (please note that hypothesis tests are always about population, but we use sample statistics to evaluate the tests), we would create null and alternative hypotheses.
Null hypothesis is the hypothesis that there is nothing going on in our population, while alternative hypothesis is the hypothesis we are testing for. Look at the above example: We have a sample of high school students, of which about 40% got into college. Then, we have someone claiming that more than 50% of high school students got accepted to college. In our hypothesis tests, we will test for the claim that more than 50% of high school students got in by setting up null and alternative hypotheses:
• Null hypothesis, denoted as H0, is p = 0.4 (i.e., 40%, the proportion of high school students accepted to college).
• Alternative hypothesis, denoted as HA, is p > 0.4 (the proportion of high school students accepted to college is higher than 0.4). Please note that even if the claim was about 50% of high school students accepted to college, 50% is still higher than 40%. We need to be consistent in the formulation of hypotheses, and thus we have p > 0.4 instead of p > 0.5 in our alternative hypothesis.
Once we conduct statistical tests, we will be able to either reject or fail to reject null hypothesis. If we gather enough evidence to reject null hypothesis, we can conclude that we have enough evidence to support alternative hypothesis. However, if we do not gather enough evidence for alternative hypothesis, we fail to reject null hypothesis. Failure to reject null hypothesis does not mean that we have to accept this hypothesis; it only means that we do not have enough evidence to reject it.
Two Types of Errors
In statistics, there are two types of errors associated with hypothesis tests. Type I error, called false positive, occurs when someone incorrectly rejects null hypothesis. For instance, we might have accidentally found some evidence to support the alternative hypothesis described in the previous section. Thus, we would claim that the proportion of high school seniors already accepted to college is higher than 40%, while it is actually not true.
Type II error, called false negative, occurs when someone incorrectly fails to reject null hypothesis. In our example of high school seniors, it would be mean that the proportion of high school seniors admitted to college is actually higher than 40%, but we did not find enough evidence to support such claim.
That’s it for today. Tomorrow, we will proceed to p-values, which can be used to evaluate null and alternative hypotheses.
Freakonomics: A Rogue Economist Explores the Hidden Side of Everything by Steven D. Levitt, Stephen J. Dubner
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