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More precisely, in traditional frequentist statistical hypothesis testing, the significance level of a test is the maximum probability of accidentally rejecting a true null hypothesis (a decision known as a Type I error). The significance of a result is also called its p-value; the smaller the p-value, the more significant the result is said to be.
For example, one may choose a significance level of, say, 5%, and calculate a critical value of a statistic (such as the mean) so that the probability of it exceeding that value, given the truth of the null hypothesis, would be 5%. If the actual, calculated statistic value exceeds the critical value, then it is significant "at the 5% level". Symbolically speaking, the significance level is denoted by α (alpha).
If the significance level is smaller, a value will be less likely to be more extreme than the critical value. So a result which is "significant at the 1% level" is more significant than a result which is "significant at the 5% level". However a test at the 1% level is more likely to have a Type II error than a test at the 5% level, and so will have less statistical power. In devising a hypothesis test, the tester will aim to maximize power for a given significance, but ultimately have to recognise that the best which can be achieved is likely to be a balance between significance and power, in other words between the risks of Type I and Type II errors. It is important to note that Type I error is not necessarily any worse than a Type II error, and vice versa. The severity of an error depends on each individual case.
If the alternative hypothesis is in fact true, then a sufficiently large sample size is likely to give a highly significant result, even if the difference between the null hypothesis and the alternative hypothesis is very small. The statistical significance of a result is therefore not an indication of how substantial or important the difference is.
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