Significance Levels and p – Values

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When conducting a hypothesis test, the criterion for rejecting the null hypothesis is that an observed value is so unlikely assuming that the null hypothesis is true that it must in fact be true. To decide this, each hypothesis test is associated with a significance level and the observed value must be less likely than the significance leveltypically 1% or 5%. When the null hypothesis is rejected, the result is said to be statistically significant.

If the distribution is a continuous distribution, then the significance level is the level of the test. A problem arises for discrete distributions, because it is usually impossible to obtain a significance level of exactly 1% or 5% or whatever level is required. For example, suppose we assume a binomial distributionand we are required to conduct a hypothesis test at the 10% level. We require the probabilities contained in the upper and lower tails to be 5% each. From the cumulative binomial distribution tables the lower end gives

and

We choose the greatest value less than 0.05 ie 0.0416.

At the upper end,andWe choose the first, this being that closest to but less than 0.05.

The total significance level is then 0.0416+0.048=0.0896.

On the other hand the– value is the probability of observing a value at least as unlike as one that is actually observed, assuming the null hypothesis is true.. For example, suppose we assume the distributionas before, and we observe 10 successes. Then the– value is

The significance level for a test conducted assuming a continuous distribution is always at least equal to the significance level of a test conducted using a discrete distribution.

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