Hypothesis Testing


TS Contributor
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Null and Alternative Hypotheses
Most hypothesis testing situations start with a null hypothesis which makes a statement about the lack of a significant difference or relationship between or among two or more parameters or variables. The researcher then declares an alternative hypothesis which is in effect, the opposite of the null, stating that there is a difference or a relationship.

For example, let's say we want to answer a research question such as:
Does car make/model X get 30 miles per gallon of gasoline in highway driving?

From this question, what we would like to find out is if car make/model x gets, on average, in the entire population, 30 miles-per-gallon of gasoline. In order to find out, we need to draw a sample from the population, examine the data from the sample, and decide whether or not the data supports the null hypothesis.

Level of Significance
The level of significance is indicative of how much evidence you, as the researcher, require before you are willing to reject the null hypothesis in favor of the alternative. Often this is expressed as an "alpha" which is a probability.

The smaller the alpha, the stronger the evidence needs to be in order to reject Ho. In other words, a small alpha would indicate that you are looking for evidence from the sample that shows that the average miles-per-gallon is much different from 30. A larger alpha would indicate that you don't require very strong evidence in order to reject Ho.

When we collect the data and run the statistical test, the results of the test include what is called a "p-value" - a probability that is directly compared to the alpha. What this probability represents is the probability that we obtained our results - the difference or relationship, or something larger or stronger, exhibited in our sample data, or obtained test statistic, if in fact the null hypothesis is true.

If the obtained p-value is less than or equal to alpha, then we have enough evidence to reject the null hypothesis - the evidence, according to our own criteria, is strong enough to dismiss Ho as "unlikely."

For instance, let's suppose that, in reality, the population of cars do in fact get 30 miles per gallon on average. If this is true, then if we draw a random sample from the population of cars that get 30 mpg, then it would be difficult to get a sample average that is much different from 30. Yes, it will likely be different from 30, but it will be somewhat close.

Most of the time, textbooks show examples where the significance level (alpha) is set at .05. At first glance, this may seem rather small, but I'll use an extreme example to show that it's not:

If you are about to board an airplane and the ticketing agent announces that there is a 5% chance that the plane will experience a catastrophic crash with no survivors, would you get on it? Get my point?

Two-Tail Tests (Non-Directional Tests)
Ho: mu = 30
Ha: mu <> 30
In this situation, we just want to see if the evidence shows that the car's mpg is different from 30. No direction is specified - it can be either larger or smaller - we don't care - we just want to know if it's different.

One-Tail Tests (Directional Tests)
Ho: mu = 30
Ha: mu > 30
In this situation, we want to see if the car's mpg is larger than 30.

Ho: mu = 30
Ha: mu < 30
In this situation, we want to see if the car's mpg is smaller than 30.

All other things being equal, directional tests are more statistically "powerful" than non-directional tests. That is, there is a greater chance of rejecting Ho in a directional test than there is in a non-directional test. The reason is that in a directional test, the rejection region is all contained in one tail of the distribution, and in a non-directional test, the rejection region is divided between the two tails. Therefore, in a nondirectional test, the test statistic would need to be larger than in a directional test in order to fall into the rejection region.

Large Samples
Z test

Let's test our hypotheses when we are able to draw a large sample from the population.

Small Samples
t test

Now let's test our hypotheses when we can only draw a small sample from the population.