Question

A certain university has decided to introduce the use of plus and minus with letter grades, as long as there is evidence that more than \(60 \%\) of the faculty favor the change. A random sample of faculty will be selected, and the resulting data will be used to test the relevant hypothe ses. If \(\pi\) represents the true proportion of all faculty that favor a change to plus- minus grading, which of the following pair of hypotheses should the administration test: $$ H_{0}: \pi=.6 \text { versus } H_{a}: \pi<.6 $$ or $$ H_{0}: \pi=.6 \text { versus } H_{a}: \pi>.6 $$ Explain your choice.

Short answer

The correct pair of hypotheses to test by the administration would be \(H_{0}: \pi = 0.6\) versus \(H_{a}: \pi > 0.6\).

Step-by-step solution

Understand the context

The decision of the university to introduce a change in the grading system depends on the faculty's opinion. The decision will only go through if more than 60% of the faculty favor it.

Formulate the hypotheses

The null hypothesis \(H_{0}\) generally assumes that the status quo is maintained, that is no change or difference exists, while the alternative hypothesis \(H_{a}\) assumes a change, a difference, or an inequality. Knowing the university's rule that change will only be implemented if more than 60% of faculty favor the change, the null hypothesis should hence be set up as \(H_{0}: \pi = 0.6\), i.e., exactly 60% of the faculty are in favor. The alternative hypothesis then is \(H_{a}: \pi > 0.6\), i.e., more than 60% of the faculty are in favor.

Choose the correct pair of hypotheses

Considering the decision rule given by the university and the interpretations of null and alternative hypotheses, the correct pair of hypotheses to test would be: \(H_{0}: \pi = 0.6\) versus \(H_{a}: \pi > 0.6\).