Mathematical Symbols For Null And Alternative Hypothesis Statements

Null Hypothesis (H0)

In many cases the purpose of research is to answer a question or test a prediction, generally stated in the form of hypotheses (-is, singular form) -- testable propositions. Examples:

Does a training program in driver safety result in a decline in accident rate? People who take a driver safety course will have a lower accident rate than those who do not take the course.
Who is better in math, men or women? Men are better at math than women.
What is the relationship between age and cell phone use? Cell phone use is higher for younger adults than for older adults.
Is there a relationship between education and income? Income increases with years of education.
Can public education reduce the occurrence of AIDS?The number of AIDS cases is inversely related to the amount of public education about the disease.

The statistical procedure for testing a hypothesis requires some understanding of the null hypothesis. Think of the outcome (dependent variable). From a statistical (and sampling) perspective), the null hypothesis asserts that the samples being compared or contrasted are drawn from the same population with regard to the outcome variable. This means that

  • any observed differences in the dependent variable (outcome) must be due to sampling error (chance)
  • the independent (predictor) variable does NOT make a difference

The symbol H0 is the abbreviation for the null hypothesis, the small zero stands for null.

Oddly enough, we are in a sense betting against our research judgment. If we didn't think that some factor made a difference, we probably would not be doing the research in the first place. But statistically speaking, we temporarily adopt the critical stance that our independent variable does NOT matter.

Generally, when comparing or contrasting groups (samples), the null hypothesis is that the difference between means (averages) = 0. For categorical data shown on a contingency table, the null hypothesis is that any differences between the observed frequencies (counts in categories) and expected frequencies are due to chance.

Research Hypothesis (H1)

The research hypothesis (or hypotheses -- there may be more than one) is our working hypothesis -- our prediction, or what we expect to happen. It is also called the alternative hypothesis - because it is an alternative to the null hypothesis. Technically, the claim of the research hypothesis is that with respect to the outcome variable, our samples are from different populations (remember that population refers to the group from which the sample is drawn). If we predict that math tutoring results in better performance, than we are predicting that after the treatment (tutoring), the treated sample truly is different from the untreated one (and therefore, from a different population).

The research or alternative hypothesis is abbreviated asH1, and if there are more hypotheses,H2,H3,H4, etc.

Why the Null Hypothesis (H0)?

When we pose a research question, we want to know whether the outcome is due to the treatment (independent variable) or due to chance (in which case our treatment is probably not effective). For example, the claim that tutoring improves math performance generally does not predict exactly how much improvement. Each level of improvement has a different probability associated with it, and it would take a long time and a great deal of effort to specify the probability of each of the possible outcomes that would support our research hypothesis.

On the other hand, the null hypothesis is straightforward -- what is the probability that our treated and untreated samples are from the same population (that the treatment or predictor has no effect)? There is only one set of statistical probabilities -- calculation of chance effects. Instead of directly testing H1, we test H0. If we can reject H0, (and extraneous factors are under control), we can accept H1. To put it another way, the fate of the research hypothesis depends upon what happens to H0.

Here are some research or alternative hypotheses (testable statements)

  • Exercise leads to weight loss
  • Exposure to classical music increases IQ score
  • Extroverts are healthier than introverts
  • Sensitivity training reduces racial bias

The inferential statistics do not directly address the testable statement (research hypothesis). They address the null hypothesis. Statistically, we test "not." Here are the null hypotheses:

  • Exercise is unrelated to weight loss.
  • Exposure to classical music has no effect on IQ score.
  • Extrovert and introverts are equally healthy.
  • People exposed to sensitivity training are no more tolerant than those not exposed to sensitivity training.

NOTE: The null hypothesis is NOT the opposite of the research hypothesis. The null hypothesis states that any effects observed after treatment (or associated with a predictor variable) are due to chance alone. Statistically, the question that is being answered is "If these samples came from the same population with regard to the outcome, how likely is the obtained result?"

Self-test #1

Next section: Introduction to Inferential statistics (testing hypotheses)

By Deborah J. Rumsey

When you set up a hypothesis test to determine the validity of a statistical claim, you need to define both a null hypothesis and an alternative hypothesis.

Typically in a hypothesis test, the claim being made is about a population parameter (one number that characterizes the entire population). Because parameters tend to be unknown quantities, everyone wants to make claims about what their values may be. For example, the claim that 25% (or 0.25) of all women have varicose veins is a claim about the proportion (that’s the parameter) of all women (that’s the population) who have varicose veins (that’s the variable — having or not having varicose veins).

Researchers often challenge claims about population parameters. You may hypothesize, for example, that the actual proportion of women who have varicose veins is lower than 0.25, based on your observations. Or you may hypothesize that due to the popularity of high heeled shoes, the proportion may be higher than 0.25. Or if you’re simply questioning whether the actual proportion is 0.25, your alternative hypothesis is: “No, it isn’t 0.25.”

How to define a null hypothesis

Every hypothesis test contains a set of two opposing statements, or hypotheses, about a population parameter. The first hypothesis is called the null hypothesis, denoted H0. The null hypothesis always states that the population parameter is equal to the claimed value. For example, if the claim is that the average time to make a name-brand ready-mix pie is five minutes, the statistical shorthand notation for the null hypothesis in this case would be as follows:

(That is, the population mean is 5 minutes.)

All null hypotheses include an equal sign in them.

How to define an alternative hypothesis

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 Ha. Here they are, along with their shorthand notations in the context of the pie example:

  • The population parameter is not equal to the claimed value

  • The population parameter is greater than the claimed value

  • The population parameter is less than the claimed value

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.

For example, if you want to test whether a company is correct in claiming its pie takes five minutes to make and it doesn’t matter whether the actual average time is more or less than that, you use the not-equal-to alternative. Your hypotheses for that test would be

If you only want to see whether the time turns out to be greater than what the company claims (that is, whether the company is falsely advertising its quick prep time), you use the greater-than alternative, and your two hypotheses are

Finally, say you work for the company marketing the pie, and you think the pie can be made in less than five minutes (and could be marketed by the company as such). The less-than alternative is the one you want, and your two hypotheses would be

How do you know which hypothesis to put in H0 and which one to put in Ha? 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 H0?


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