Final answer:
The alternative hypothesis that is not valid is 'x₁ < x₂' because alternative hypotheses involve population means (μ), not sample means (x). The valid alternative hypotheses could be 'μ₁ < μ₂', 'μ₁ ≠ μ₂', or 'μ₁ > μ₂', and the type of test used—two-tailed or one-tailed—depends on the alternative hypothesis's symbol.
Step-by-step explanation:
Given two samples for which we want to contrast the means, the alternative hypothesis that is not valid would be the statement x₁ < x₂. This is because alternative hypotheses involve population means, denoted by μ, not sample means, denoted by x. Thus, out of the options provided, μ₁ < μ₂, μ₁ ≠ μ₂, and μ₁ > μ₂ could all be valid alternative hypotheses. If the alternative hypothesis contains a not equals symbol (≠), you would use a two-tailed test. Assuming the null hypothesis states that the mean is at least 18 or that it is at most 12, you would use a one-tailed test, the direction of which depends on the nature of the inequality (left-tailed for 'at least' and right-tailed for 'at most'). If the null hypothesis states that the mean is equal to a certain number and the alternative states it is not equal, then it is a two-tailed test.