Question

Determine whether the sets of hypotheses given are valid hypotheses. State whether each set of hypotheses is valid for a statistical test. If not valid, explain why not. (a) \(H_{0}: \rho=0 \quad\) vs \(\quad H_{a}: \rho<0\) (b) \(H_{0}: \hat{p}=0.3 \quad\) vs \(\quad H_{a}: \hat{p} \neq 0.3\) (c) \(H_{0}: \mu_{1} \neq \mu_{2} \quad\) vs \(\quad H_{a}: \mu_{1}=\mu_{2}\) (d) \(H_{0}: p=25 \quad\) vs \(\quad H_{a}: p \neq 25\)

Short answer

a) Valid b) Valid c) Invalid d) Valid

Step-by-step solution

Evaluate first set

For the first set of hypothesis, we have \(H_{0}: \rho=0\) and \(H_{a}: \rho<0\). It is acceptable because if the null hypothesis is rejected, the alternative is automatically considered to be true. Thus, it's a valid set of hypotheses.

Evaluate second set

For the second set \(H_{0}: \hat{p}=0.3\) and \(H_{a}: \hat{p} \neq 0.3\), we can also observe this sort of opposition between the null hypothesis and alternative hypothesis, thus they are valid.

Evaluate third set

For hypothesis pair \(H_{0}: \mu_{1} \neq \mu_{2}\) and \(H_{a}: \mu_{1}=\mu_{2}\), the null hypothesis assumes inequality, while the alternative assumes equality. This is the opposite of the usual setup, hence it is not a valid set. Null hypothesis should always contain equal sign.

Evaluate fourth set

For the fourth and final set, \(H_{0}: p=25\) and \(H_{a}: p \neq 25\), there is a clear opposition between the null and alternative hypothesis where the null hypothesis states that the parameter is equal to a specific value, and the alternative hypothesis states that it does not equal that value. Hence, this represents a valid set of hypotheses.