Summary

• In order to run a formal hypothesis test at any level of confidence, we must relate sampling distributions and sample statistics to the standard normal distribution Z.
• If the sampling distribution of a parameter follows a normal distribution, then it can be transformed into Z using the transformation formula.
• The z-score of an observed sample statistic is then known as a test statistic.
• The test statistic gives an indication of where the sample statistic is in the appropriate sampling distribution.
• For a hypothesis test, the level of significance, α, is defined to be the probability of rejecting the null hypothesis if the null hypothesis is true.
• The level of significance is fixed by the person running the hypothesis test.
• The level of significance determines the critical values and region of rejection for the hypothesis test.
• The critical values and region of rejection will depend on whether the test is one-sided or two-sided.
• In a two-sided test with level of significance α:
  • The critical values are the two z-scores z_α/2 and -z_α/2 defined such that a proportion of α/2 of Z falls above z_α/2 and a proportion of α/2 falls below -z_α/2.
  • The region of rejection is the area of Z that falls outside of the critical values in Z. That is, it is the area below -z_α/2 and the area above z_α/2.
• In a one-sided test with level of significance α:
  • If the alternative hypothesis asserts that the population parameter is greater than the value in the null hypothesis, the critical value is the z-score z_α. If the alternative hypothesis asserts that the population parameter is less than the value in the null hypothesis, the critical value is the z-score -z_α.
  • The region of rejection is the area of Z that falls outside of the critical value in Z. That is, the area above z_α (if that is the critical value), or below -z_α (if that is the critical value).
• Once a sample is collected, the test statistic is to be calculated. If the test statistic is in the region of rejection, the null hypothesis is rejected. If the test statistic is not in the region of rejection, the null hypothesis is not rejected.