Question

Which of the following specify legitimate pairs of null and alternative hypotheses? a. \(H_{0}: p=0.25 \quad H_{a}: p>0.25\) b. \(H_{0}: p<0.40 \quad H_{a}: p>0.40\) c. \(H_{0}: p=0.40 \quad H_{a}: p<0.65\) d. \(H_{0}: p \neq 0.50 \quad H_{a}: p=0.50\) e. \(H_{0}: p=0.50 \quad H_{a}: p>0.50\) f. \(H_{0}: \hat{p}=0.25 \quad H: \hat{p}>0.25\)

Short answer

The legitimate pairs of null and alternative hypotheses are: a. \(H_{0}: p=0.25, H_{a}: p>0.25\), e. \(H_{0}: p=0.50, H_{a}: p>0.50\), and f. \(H_{0}: \hat{p}=0.25, H: \hat{p}>0.25\), as long as `p` refers to a sample proportion in the context of the problem.

Step-by-step solution

Evaluate Each Pair of Hypotheses

Each of the proposed pairs of hypotheses should be evaluated based on the criteria that the null hypothesis \(H_0\) should contain a statement of equality, and the alternative \(H_a\) should contain a statement of inequality. The hypothesis \(H_0\) should state either that the parameter equals a specified value, is less than or equal to a specified value, or is greater than or equal to a specified value, while \(H_a\) should state either that parameter is different than the specified value, is less than the specified value, or is greater than the specified value.

Scoring the Alternatives

Each alternative is evaluated in the following way: a. \(H_{0}: p=0.25 \quad H_{a}: p>0.25\) is a legitimate pair of hypotheses. b. \(H_{0}: p<0.40 \quad H_{a}: p>0.40\) is not a legitimate pair, because the null hypothesis contains an inequality. c. \(H_{0}: p=0.40 \quad H_{a}: p<0.65\) this is also incorrect because the alternative hypothesis contains a partial equal sign (<=). d. \(H_{0}: p \neq 0.50 \quad H_{a}: p=0.50\) is not a legitimate pair as well because the null hypothesis contains inequality and alternative contains an equality. e. \(H_{0}: p=0.50 \quad H_{a}: p>0.50\) is a legitimate pair of hypotheses. f. \(H_{0}: \hat{p}=0.25 \quad H: \hat{p}>0.25\) is a legitimate pair of hypotheses, as long as `p` refers to a sample proportion in the context of the problem.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. The process involves posing two contrasting statements: the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\) or \(H_1\). These hypotheses are formulated before any data is collected.

The null hypothesis is a statement that indicates no effect or no difference and is assumed to be true until evidence suggests otherwise. On the other hand, the alternative hypothesis is what a researcher wants to prove—indicating some effect or difference. In hypothesis testing, statistical evidence is used to determine whether there is enough support to reject the null hypothesis in favor of the alternative hypothesis.

The outcome of hypothesis testing is either to reject the null hypothesis (if the test statistic falls within the critical region or the p-value is less than the significance level) or to fail to reject it (if the test statistic does not fall within the critical region or the p-value is greater than the significance level). It's important to note that failing to reject the null hypothesis does not prove it true; it simply means there is not enough evidence against it based on the sample data.

Statistical Hypotheses

Statistical hypotheses are precise statements about population parameters that can be tested statistically. They are the foundation for hypothesis testing. There are two types of statistical hypotheses: the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\).

The null hypothesis \(H_0\) often includes statements of equality such as \(p = k\) where \(p\) is a population parameter and \(k\) is a specific value. Alternatively, it could consist of statements indicating no smaller than or no larger than a certain value, for instance \(p \geq k\) or \(p \leq k\). Its role is to be the default or status quo condition that the test aims to challenge.

The alternative hypothesis \(H_a\) conversely, suggests inequality such as \(p eq k\), \(p > k\), or \(p < k\). It represents all the other possible conditions contradictory to the null hypothesis and encompasses the claim the test seeks to support with evidence. For example, if a researcher wants to prove that a new medication is more effective than the current standard, the alternative hypothesis would state that the effectiveness of the new medication is greater than that of the standard.

Inequality in Hypotheses

Inequality in hypotheses is a key aspect to understanding which hypothesis contains an equality statement and which contains an inequality. The null hypothesis \(H_0\) should always be a statement of equality or non-strict inequality (\(\geq\), \(\leq\)), signifying no change or effect. This statement establishes a baseline for comparison. The expectation is that unless the data provides convincing evidence to the contrary, the null hypothesis is presumed to hold true.

The alternative hypothesis \(H_a\), in contrast, is a statement of strict inequality (\(eq\), \(>\), \(<\)), representing a departure from the null hypothesis. When setting up hypothesis tests, care must be taken to correctly formulate these inequalities. Mixed directional statements, like having \(H_0\) state a ‘less than’ inequality and \(H_a\) a ‘greater than’ inequality, are invalid because they fail to be mutually exclusive and collectively exhaustive. This means that one hypothesis should cover all possibilities not covered by the other so that between them, all possible outcomes are accounted for without overlap.

When reviewing hypothesis pairs, as in the original exercise, it is crucial to ensure that \(H_0\) contains an equality or non-strict inequality and that \(H_a\) includes a strict inequality that stands as the logical opposite of \(H_0\). This is essential for the integrity of the hypothesis test and to accurately interpret the results.