Question

Explain why the statement \(\hat{p}>0.50\) is not a legitimate hypothesis.

Short answer

The statement \(\hat{p}>0.50\) is not a legitimate hypothesis because it's not an exact statement about a population parameter; rather, it's an inequality. Legitimate hypotheses should equate to a specific value, not contain inequalities.

Step-by-step solution

Understanding Hypotheses

In statistical hypothesis testing, a hypothesis is an assumption made about a population parameter. There are two types of statistical hypotheses: Null hypothesis (H0): The null hypothesis asserts that the population parameter equals a particular value. Alternative hypothesis (H1): The alternative hypothesis contradicts the null hypothesis, asserting any other possible conditions other than that which is stated in H0. The null hypothesis and alternative hypothesis must account for all possible outcomes of the population parameter.

Identifying the Problem with the Given Statement

Looking at the given statement \(\hat{p}>0.50\), it seems that it's intended to be a hypothesis for a proportion, with the symbol \(\hat{p}\) typically representing a sample proportion. The problem here is the mention of a 'greater than' symbol. Hypotheses should not contain inequalities; they should be clear, concise and exact. They typically equate the population parameter to a specific value. Therefore, the statement \(\hat{p}>0.50\) is not a clear-cut hypothesis.

Solution Explanation

Since the hypotheses should not contain inequalities but should equate to a specific value, the statement \(\hat{p}>0.50\) does not constitute a legitimate hypothesis. A correct null hypothesis might be H0: \(\hat{p}=0.50\), and a corresponding alternative could be H1: \(\hat{p}\neq 0.50\). These hypotheses encompass all possible conditions - p is either 0.50, or it's not - thus meeting the criteria for legitimate hypotheses.