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A statistical hypothesis test is a method of making decisions using data from a scientific study.
Outline the steps of a standard hypothesis test.
Statistical hypothesis tests define a procedure that controls (fixes) the probability of incorrectly deciding that a default position (null hypothesis) is incorrect based on how likely it would be for a set of observations to occur if the null hypothesis were true.
The first step in a hypothesis test is to state the relevant null and alternative hypotheses; the second is to consider the statistical assumptions being made about the sample in doing the test.
Next, the relevant test statistic is stated, and its distribution is derived under the null hypothesis from the assumptions.
After that, the relevant significance level and critical region are determined.
Finally, values of the test statistic are observed and the decision is made whether to either reject the null hypothesis in favor of the alternative or not reject it.
A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical evidence in the form of a hypothesis test indicates otherwise.
In a famous example of hypothesis testing, known as the Lady tasting tea example, a female colleague of Sir Ronald Fisher claimed to be able to tell whether the tea or the milk was added first to a cup. Fisher proposed to give her eight cups, four of each variety, in random order. One could then ask what the probability was for her getting the number she got correct, but just by chance. The null hypothesis was that the Lady had no such ability. The test statistic was a simple count of the number of successes in selecting the 4 cups. The critical region was the single case of 4 successes of 4 possible based on a conventional probability criterion (< 5%; 1 of 70 ≈ 1.4%). Fisher asserted that no alternative hypothesis was (ever) required. The lady correctly identified every cup, which would be considered a statistically significant result.
A statistical hypothesis test is a method of making decisions using data from a scientific study. In statistics, a result is called statistically significant if it has been predicted as unlikely to have occurred by chance alone, according to a pre-determined threshold probability—the significance level. Statistical hypothesis testing is sometimes called confirmatory data analysis, in contrast to exploratory data analysis, which may not have pre-specified hypotheses. Statistical hypothesis testing is a key technique of frequentist inference.
Statistical hypothesis tests define a procedure that controls (fixes) the probability of incorrectly deciding that a default position (null hypothesis) is incorrect based on how likely it would be for a set of observations to occur if the null hypothesis were true. Note that this probability of making an incorrect decision is not the probability that the null hypothesis is true, nor whether any specific alternative hypothesis is true. This contrasts with other possible techniques of decision theory in which the null and alternative hypothesis are treated on a more equal basis.
The Testing Process
The typical line of reasoning in a hypothesis test is as follows:
There is an initial research hypothesis of which the truth is unknown.
The first step is to state the relevant null and alternative hypotheses. This is important as mis-stating the hypotheses will muddy the rest of the process.
The second step is to consider the statistical assumptions being made about the sample in doing the test—for example, assumptions about the statistical independence or about the form of the distributions of the observations. This is important because invalid assumptions will mean that the results of the test are invalid.
Decide which test is appropriate, and state the relevant test statisticT.
Derive the distribution of the test statistic under the null hypothesis from the assumptions.
Select a significance level (α), a probability threshold below which the null hypothesis will be rejected. Common values are 5% and 1%.
The distribution of the test statistic under the null hypothesis partitions the possible values of T into those for which the null hypothesis is rejected, the so called critical region, and those for which it is not. The probability of the critical region is α.
Compute from the observations the observed value tobs of the test statistic T.
Decide to either reject the null hypothesis in favor of the alternative or not reject it. The decision rule is to reject the null hypothesis H0 if the observed value tobs is in the critical region, and to accept or "fail to reject" the hypothesis otherwise.
An alternative process is commonly used:
7. Compute from the observations the observed value tobs of the test statistic T.
8. From the statistic calculate a probability of the observation under the null hypothesis (the p-value).
9. Reject the null hypothesis in favor of the alternative or not reject it. The decision rule is to reject the null hypothesis if and only if the p-value is less than the significance level (the selected probability) threshold.
The two processes are equivalent. The former process was advantageous in the past when only tables of test statistics at common probability thresholds were available. It allowed a decision to be made without the calculation of a probability. It was adequate for classwork and for operational use, but it was deficient for reporting results. The latter process relied on extensive tables or on computational support not always available. The calculations are now trivially performed with appropriate software.
Tea Tasting Distribution
This table shows the distribution of permutations in our tea tasting example.
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