Question

The null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1} \neq \mu_{2}\) Sample: \(\bar{x}_{1}=2.7\) and \(\bar{x}_{2}=2.1\) Randomization statistic \(=\bar{x}_{1}-\bar{x}_{2}\)

Short answer

The randomization distribution would be centered at 0, and the test would be a two-tailed test.

Step-by-step solution

Interpreting the Hypotheses

The null hypothesis (H0) asserts that the means of population 1 and 2 are equal, denoted by \(H_{0}: \mu_{1}=\mu_{2}\). The alternative hypothesis (Ha), on the other hand, contends that the means are not equal, denoted by \(H_{a}: \mu_{1} \neq \mu_{2}\). Given that Ha involves a 'not equals to' condition, it suggests a two-tailed test.

Understanding the Sample Means

The given sample means are \(\bar{x}_{1}=2.7\) and \(\bar{x}_{2}=2.1\). Thus, the mean of sample 1 is greater than that of sample 2. This information will be used in the next step.

Calculating Randomization Statistic and Predicting the Center

The randomization statistic is given by \(=\bar{x}_{1}-\bar{x}_{2}\), which translates to \(=2.7-2.1=0.6\). However, under the null hypothesis (H0), the two population means are assumed to be equal so the expected difference between them would be zero. Consequently, the center of the randomization distribution should be at 0.