Question

A college has decided to introduce the use of plus and minus with letter grades, as long as there is convincing 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 hypotheses. If \(p\) represents the proportion of all faculty who favor a change to plus-minus grading, which of the following pairs of hypotheses should be tested? $$H_{0}: p=0.6 \text { versus } H_{a}: p<0.6$$ or $$H_{0}: p=0.6 \text { versus } H_{a}: p>0.6$$ Explain your choice.

Short answer

The correct pair of hypotheses to be tested is \(H_{0}: p=0.6\) versus \(H_{a}: p>0.6\). The null hypothesis posits that the proportion of faculty in favor of the grading change is 60%, and the alternative hypothesis suggests that this proportion is greater than 60%.

Step-by-step solution

Identify the correct null hypothesis

The null hypothesis, \(H_{0}\), is a statement of no effect or no difference. It is the hypothesis that we want to provide evidence against. In this case, the status quo is that 60% of the faculty favor the change. So the null hypothesis (\(H_{0}\)) should be \( p = 0.6 \).

Identify the correct alternate hypothesis

The alternative hypothesis, \(H_{a}\), is the hypothesis that we will accept if we find enough evidence against the null hypothesis. The college wants to introduce the change if there's convincing evidence that MORE than 60% of the faculty favor the change. So the alternate hypothesis (\(H_{a}\)) should be \( p > 0.6 \).

Key concepts

Null Hypothesis

The null hypothesis (\textbf{H\textsubscript{0}}) is a fundamental aspect of hypothesis testing in statistics. Think of it as the default statement about a population parameter that we presume to be true until statistical evidence suggests otherwise. It often reflects a situation of no change, no effect, or no difference.

In the context of the college wanting to change the grading system, the null hypothesis is that the proportion of faculty members who support the new grading system, denoted by \( p \), is 60% (\textbf{H\textsubscript{0}}: \( p = 0.6 \)). Since the college aims to adopt the plus-minus grading if convincing evidence that more than 60% of faculty are in favor exists, the starting assumption or null hypothesis must reflect the current belief or status quo before we analyze the sample data.

Alternative Hypothesis

The alternative hypothesis (\textbf{H\textsubscript{a}}) represents what we suspect might actually be true and what we're trying to find evidence for. Unlike the null hypothesis, the alternative hypothesis points to a new effect, a difference, or a change from the status quo.

In our grading system example, the college is interested in whether a greater proportion than the assumed 60% favor the change (\textbf{H\textsubscript{0}}: \( p = 0.6 \)). Thus, the alternative hypothesis reflects this possibility and is stated as \textbf{H\textsubscript{a}}: \( p > 0.6 \). If our statistical test provides sufficient evidence to support \textbf{H\textsubscript{a}}, the college may then decide to implement the grading system change.

Proportions

In statistics, proportions represent a part of a whole, typically expressed as a percentage or a fraction. In hypothesis testing, we're often interested in the proportion of a certain characteristic within a population. For instance, the proportion of faculty members who favor a new grading policy could influence decision-making.

The problem provided focuses on the proportion, denoted as \( p \), of all faculty who favor a change to plus-minus grading. Proportions are essential as they serve as the parameter we're testing, and differentiating between proportions greater than, less than, or equal to a specified value can alter the direction and interpretation of a hypothesis test.

Statistical Evidence

Statistical evidence refers to the data analysis results obtained from a sample that can either support or refute a given hypothesis. The evidence is typically quantified using a p-value, which measures the strength of the evidence against the null hypothesis. A low p-value indicates that it is unlikely to observe such data if the null hypothesis is true, leading researchers to consider the alternative hypothesis.

In the exercise, convincing statistical evidence would be data that shows more than 60% of the sample faculty favor the change. The decision to reject or not reject the null hypothesis will be based on this evidence, which comes from calculating p-values and confidence intervals in the context of the sample data gathered.