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For an inferential statistic such as a one-sided t, an F or a chi-square test, a critical value is the number above which a fraction of the values of the inference statistics equal to the alpha level would fall on repeated trials if the null hypothesis were true.

For example, suppose the research has chosen an alpha level of 0.05. She has a sample size of 11 and will be using a one-sided t-statistic because she is interested in deciding whether the mean of the population from which she has drawn her sample exceeds a certain given value. The critical value for a t-test in this situation is about 1.8 because about 0.05 of the time anyone could take a sample of size 11 from a population with a known mean and find that the t-statistic calculated for the sample exceeds 1.8.

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Q: What is the critical value of an inferential statistic and how does it relate to the alpha level?
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