4.9 Setting Up a Test for the Difference Between Two Population Means

Identify an appropriate testing method for the difference between two population means including the parameters for the difference between the two population means.

• The appropriate test for the difference between two population means is a twosample t-test for a difference between two population means.
• The parameters for a hypothesis test for the difference between two population means should reference the population parameters, the response variables, and the populations in context.

Identify the null and alternative hypotheses for the difference between two population means.

• The null hypothesis for a two-sample t-test for the difference between two population means, µ1 and µ2 , can be written as either H:01 μμ-= 2 0 or H:01 μμ= 2 . A one-sided alternative hypothesis for the difference between population means can be written as either H:a1 μμ< 2 (or equivalently H:a μμ12-< 0 or H:a1 μμ> 2 (or equivalently H:a μμ12-> 0). A two-sided alternative hypothesis for the difference between population means can be written as H:a μμ12≠ (or equivalently H:a μμ12-≠ 0).

Justify the appropriateness of a hypothesis test for the difference between two population means by verifying conditions.

• A two-sample t-test for a difference between population means requires that three conditions be met:
  • i. The randomization condition—the data should be collected using two independent random samples or a randomized experiment.
  • ii. The 10% condition—when sampling without replacement, the size of each sample should be less than or equal to 10% of the respective population size: n11≤10%N and n22≤10%N , where N1 is the size of population 1 and N2 is the size of population 2. The sample sizes are represented as n1 and n2. (Note: This condition is unnecessary when the data are from a randomized experiment.)
  • iii. The sample data condition—both samples should have a sample size greater than or equal to 30 or it is indicated that both population distributions are approximately normal. If either sample size is less than 30, both sample data distributions should be free from strong skewness and outliers. 139 Inference for Quantitative Data: Means UNIT 4