Example 11.2. Hypotheses with One Sample of One Categorical Variable
Null and Alternative Hypotheses for a Mean
State Null and Alternative Hypotheses
This module will continue the discussion of hypothesis testing, where a specific statement or hypothesis is generated about a population parameter, and sample statistics are used to assess the likelihood that the hypothesis is true. The hypothesis is based on available information and the investigator's belief about the population parameters. The specific test considered here is called analysis of variance (ANOVA) and is a test of hypothesis that is appropriate to compare means of a continuous variable in two or more independent comparison groups. For example, in some clinical trials there are more than two comparison groups. In a clinical trial to evaluate a new medication for asthma, investigators might compare an experimental medication to a placebo and to a standard treatment (i.e., a medication currently being used). In an observational study such as the Framingham Heart Study, it might be of interest to compare mean blood pressure or mean cholesterol levels in persons who are underweight, normal weight, overweight and obese.
A null hypothesis (H0) exists when a researcher believes there is no relationship between the two variables, or there is a lack of information to state a scientific hypothesis. This is something to attempt to disprove or discredit.
State Null and Alternative Hypotheses
A fairly common criticism of the hypothesistesting approach to statistics is that the null hypothesis will always be false, if you have a big enough sample size. In the chickenfeet example, critics would argue that if you had an infinite sample size, it is impossible that male chickens would have exactly the same average foot size as female chickens. Therefore, since you know before doing the experiment that the null hypothesis is false, there's no point in testing it.
How do you know which hypothesis to put in H_{0} and which one to put in H_{a}? Typically, the null hypothesis says that nothing new is happening; the previous result is the same now as it was before, or the groups have the same average (their difference is equal to zero). In general, you assume that people’s claims are true until proven otherwise. So the question becomes: Can you prove otherwise? In other words, can you show sufficient evidence to reject H_{0}?
State Null and Alternative Hypotheses
Note that this research question might also be addressed like example 11.4 by making the hypotheses about comparing the proportion of stroke patients that live with smokers to the proportion of controls that live with smokers.
Usually, the null hypothesis is boring and the alternative hypothesis is interesting. For example, let's say you feed chocolate to a bunch of chickens, then look at the sex ratio in their offspring. If you get more females than males, it would be a tremendously exciting discovery: it would be a fundamental discovery about the mechanism of sex determination, female chickens are more valuable than male chickens in egglaying breeds, and you'd be able to publish your result in Science or Nature. Lots of people have spent a lot of time and money trying to change the sex ratio in chickens, and if you're successful, you'll be rich and famous. But if the chocolate doesn't change the sex ratio, it would be an extremely boring result, and you'd have a hard time getting it published in the Eastern Delaware Journal of Chickenology. It's therefore tempting to look for patterns in your data that support the exciting alternative hypothesis. For example, you might look at 48 offspring of chocolatefed chickens and see 31 females and only 17 males. This looks promising, but before you get all happy and start buying formal wear for the Nobel Prize ceremony, you need to ask "What's the probability of getting a deviation from the null expectation that large, just by chance, if the boring null hypothesis is really true?" Only when that probability is low can you reject the null hypothesis. The goal of statistical hypothesis testing is to estimate the probability of getting your observed results under the null hypothesis.
Null hypothesis: μ = 72 Alternative hypothesis: μ ≠72

State Null and Alternative Hypotheses
Example 11.6. Hypotheses about the relationship between Two Categorical Variables

State Null and Alternative Hypotheses
Example 11.7. Hypotheses about the relationship between Two Measurement Variables

Null hypothesis: μ = 72 Alternative hypothesis: μ ≠72
Example 11.8. Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples
Example of a complex multiple independent variable hypothesis:
The probability that was calculated above, 0.030, is the probability of getting 17 or fewer males out of 48. It would be significant, using the conventional PP=0.03 value found by adding the probabilities of getting 17 or fewer males. This is called a onetailed probability, because you are adding the probabilities in only one tail of the distribution shown in the figure. However, if your null hypothesis is "The proportion of males is 0.5", then your alternative hypothesis is "The proportion of males is different from 0.5." In that case, you should add the probability of getting 17 or fewer females to the probability of getting 17 or fewer males. This is called a twotailed probability. If you do that with the chicken result, you get P=0.06, which is not quite significant.
State Null and Alternative Hypotheses
Before actually conducting a hypothesis test, you have to put two possible hypotheses on the table — the null hypothesis is one of them. But, if the null hypothesis is rejected (that is, there was sufficient evidence against it), what’s your alternative going to be? Actually, three possibilities exist for the second (or alternative) hypothesis, denoted H_{a}. Here they are, along with their shorthand notations in the context of the pie example:
Example of a complex multiple dependent variable hypothesis:
Which alternative hypothesis you choose in setting up your hypothesis test depends on what you’re interested in concluding, should you have enough evidence to refute the null hypothesis (the claim). The alternative hypothesis should be decided upon before collecting or looking at any data, so as not to influence the results.
How to Set Up a Hypothesis Test: Null versus Alternative
Every hypothesis test contains a set of two opposing statements, or hypotheses, about a population parameter. The first hypothesis is called the denoted H_{0}. The null hypothesis always states that the population parameter is to the claimed value. For example, if the claim is that the average time to make a namebrand readymix pie is five minutes, the statistical shorthand notation for the null hypothesis in this case would be as follows: