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Overview of Frequentist Hypothesis Testing

Most commonly-used frequentist hypothesis tests involve the following elements:
  1. Model assumptions (e.g., for the t-test for the mean, the model assumptions can be phrased as: simple random sample1 of a random variable with a normal distribution)
  2. Null and alternative hypothesis
  3. A test statistic. This needs to have the property that extreme values of the test statistic cast doubt on the null hypothesis.
  4. A mathematical theorem saying, "If the model assumptions and the null hypothesis are both true, then the sampling distribution of the test statistic has this particular form."2

The exact details of these four elements will depend on the particular hypothesis test. We will illustrate with an example.


In the case of the large-sample z-test for the mean, the elements are:

1. Model assumptions: We are dealing with simple random samples of the random variable X  which has a normal distribution.1

2. Null hypothesis: The mean of the random variable in question is a certain value µ0. The alternative hypothesis could be either "The mean of the random variable X is not µ0," or "The mean of the random variable X is less than µ0," or "The mean of the random variable X is greater than µ0." For this example, we will use the first alternative, "The mean of the random variable is not µ0." (This is called the two-sided alternative.)

3. Test statistic: x-bar

We now  step back and consider all possible simple random samples of X of size n. For each simple random sample of X of size n, we get a value of x-bar. We thus have a new random variable X-bar. (X-bar stands for the new random variable; x-bar stands for the value of X-bar for a particular sample of size n.) The distribution of X-bar is called the sampling distribution of X-bar.

4. The theorem states: If the model assumptions are true and if the mean of X is µ0, then the sampling distribution is normal, with mean µ0 and standard deviation σ/(√n), where σ (sigma) is the standard deviation of the random variable X. (Note: σ is called the population standard deviation of X; it is not the same as the sample standard deviation s, although s is an estimate of σ.)

The validity of the hypothesis test depends on the truth of the conclusion of the theorem; the only way we know the conclusion is true is if we know the  hypotheses of the theorem are true. Thus: If the model assumptions are not true, then we do not know that the theorem is true, so we do not know that the hypothesis test is valid.

In the example , this translates to: If the sample is not a simple random sample, then the reasoning establishing the validity of the hypothesis test breaks down.

  1. Different hypothesis tests have different model assumptions. Some tests apply to random samples that are not simple; see Other Types of Random Samples. For many tests, the model assumptions consist of several assumptions. If any one of these model assumptions is not true, we do not know that the test is valid. 
  2. Many techniques are robust to departures from at least some model assumptions. This means that if the particular assumption is not too far from true, then the technique is still approximately valid

1. This refers to a simple random sample of a random variable; see the page More Precise Definition of Sample Random Sample for more information.
2. The distribution of the test statistic, when considering all possible suitably random samples of the same size, is called a sampling distribution. For additional discussion of sampling distributions, see 
Overview of Frequentist Confidence Intervals and Frequentist Hypothesis Tests, p-values, and Type I Error. Those two pages and this one are best read as a unit.

Last modified May 10, 2012