3.6 p-Values

Interpret the p-value of a hypothesis test for a population proportion.

• Given the null hypothesis is true, there is a probability distribution of the test statistic called the null distribution. Using the null distribution, the p-value is the probability of obtaining a test statistic as extreme or more extreme (i.e., in the direction of the alternative hypothesis) than the test statistic that is observed given that the null hypothesis is true. That is, when x is the test statistic, the p-value is determined by finding the following:
  • i. The probability at or above the observed value of the test statistic Pz x , if the alternative is >
  • ii. The probability at or below the observed value of the test statistic Pz x , if the alternative is <
  • iii. The probability less than or equal to the negative of the absolute value of the test statistic plus the probability greater than or equal to the absolute value of the test Pz xP z x ,statistic, if the alternative is ≠
• If the distribution of the test statistic has been simulated, the p-value is the proportion of values in the null distribution that are as extreme or more extreme than the observed value of the test statistic. This is as follows:
  • i. The proportion at or above the observed value of the test statistic, if the alternative is >
  • ii. The proportion at or below the observed value of the test statistic, if the alternative is <
  • iii. The proportion less than or equal to the negative of the absolute value of the test statistic plus the proportion greater than or equal to the absolute value of the test statistic, if the alternative is
• An interpretation of the p-value of a hypothesis test for a population proportion should include a statement that the p-value is computed by assuming the null hypothesis is true (i.e., by assuming the true population proportion is equal to the particular value stated in the null hypothesis in context).
• Small p-values indicate that the observed value of the test statistic would be unusual if the null hypothesis were true and therefore provide evidence for the alternative hypothesis. The lower the p-value, the more convincing the statistical evidence for the alternative hypothesis.
• p-values that are not small indicate that the observed value of the test statistic would not be unusual if the null hypothesis were true and therefore do not provide convincing statistical evidence for the alternative hypothesis, nor do they provide evidence that the null hypothesis is true. Inference for Categorical Data: Proportions UNIT 3 96