In hypothesis testing, what are the two types of hypotheses that are formulated, and how are they related to each other?

• A. The descriptive hypothesis and the inferential hypothesis; they are used to summarize and analyze data, respectively.
• B. The sample hypothesis and the population hypothesis; they are both tested independently.
• C. The $\mathrm{null}$ hypothesis and the $\mathrm{alternative}$ hypothesis; they are mutually exclusive and collectively exhaustive statements about a population parameter.
• D. The one-tailed hypothesis and the two-tailed hypothesis; they are both accepted if the $p$-value is large.

Answer: (C)

Verified step by step guidance

1. Understand that in hypothesis testing, we formulate two competing hypotheses about a population parameter to make a decision based on sample data.
2. The first hypothesis is called the null hypothesis, denoted as \(H_0\), which represents a statement of no effect or no difference. It is the default assumption that there is nothing new or unusual happening.
3. The second hypothesis is called the alternative hypothesis, denoted as \(H_1\) or \(H_a\), which represents a statement that contradicts the null hypothesis. It reflects the presence of an effect or difference.
4. These two hypotheses are mutually exclusive, meaning they cannot both be true at the same time, and collectively exhaustive, meaning one of them must be true.
5. The goal of hypothesis testing is to use sample data to decide whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis.