# LIS 504 - Critical values

A common way of testing a hypothesis is to ask how likely the results we see are, if the hypothesis is true. A hypothesis often predicts a particular value for a measure, plus or minus some difference for error. The farther the observed value is from the predicted, the less likely it is to be observed if the hypothesis is true. The following chart shows the probability of getting a value for a sample measure at least a given distance from the expected value if the sample measures follow a normal distribution.
 Probability Distance from expected value (in standard errors)

We can see, for instance, that, if the observed value is 2 standard errors from the expected value, the probability of getting a value at least that different if the hypothesis is true is about 0.05, or 5%.

A critical value is a value determined in advance to decide whether a hypothesis will be accepted or rejected. If an observed value is at or beyond the critical value (in the rejection region), the hypothesis is rejected; otherwise (if the observed value is in the acceptance region), the hypothesis is accepted. In the chart above, if the critical value had been set at 2 standard errors, there would be about a 5% chance, if the hypothesis were true, that it would be rejected.

If the hypothesis is the null hypothesis, rejecting it when it is true is called a type-I error. If the hypothesis in the chart is the null hypothesis, the probability of a type-I error if the critical value is set at 2 standard errors is about 5%.

Accepting the null hypothesis when it is false is called a type-II error. The probability of a type-II error is hard to estimate, because there are so many different ways in which a hypothesis could be false. Generally, however, the probability of a type-II error is less for larger samples than for smaller samples.

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Last updated November 1, 2000.