$H_a$

Definition

$H_a$ is the alternative hypothesis in a statistical hypothesis test. It represents the statement that the researcher believes to be true, in contrast to the null hypothesis ($H_0$), which is the statement that the researcher is trying to disprove. The alternative hypothesis is the hypothesis that the researcher hopes to accept if the null hypothesis is rejected.

5 Must Know Facts For Your Next Test

1. The alternative hypothesis, $H_a$, is the hypothesis that the researcher believes to be true and wants to provide evidence for.
2. In a hypothesis test, the null hypothesis ($H_0$) and the alternative hypothesis ($H_a$) are mutually exclusive, meaning that only one of them can be true.
3. The alternative hypothesis can take different forms, such as a one-sided hypothesis (e.g., $H_a: \\mu > \\mu_0$) or a two-sided hypothesis (e.g., $H_a: \\mu \neq \\mu_0$).
4. The decision to reject or fail to reject the null hypothesis is based on the p-value, which is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.
5. The alternative hypothesis plays a crucial role in determining the appropriate test statistic and the direction of the rejection region in a hypothesis test.

Review Questions

• Explain the relationship between the null hypothesis ($H_0$) and the alternative hypothesis ($H_a$) in a hypothesis test.
  • The null hypothesis ($H_0$) and the alternative hypothesis ($H_a$) are mutually exclusive statements in a hypothesis test. The null hypothesis represents the assumption or claim that the researcher is trying to disprove, while the alternative hypothesis is the statement that the researcher believes to be true and hopes to provide evidence for. The outcome of the hypothesis test will either lead to the rejection of the null hypothesis in favor of the alternative hypothesis, or the failure to reject the null hypothesis, which means there is not enough evidence to support the alternative hypothesis.
• Describe the different forms that the alternative hypothesis ($H_a$) can take and how they affect the hypothesis test.
  • The alternative hypothesis ($H_a$) can take different forms, depending on the research question and the type of hypothesis test being conducted. One-sided alternative hypotheses, such as $H_a: \\mu > \\mu_0$, specify the direction of the difference or relationship being tested. Two-sided alternative hypotheses, such as $H_a: \\mu \neq \\mu_0$, do not specify a direction and simply indicate that there is a difference between the parameters. The form of the alternative hypothesis affects the test statistic used, the rejection region, and the interpretation of the p-value in the hypothesis test.
• Explain the role of the alternative hypothesis ($H_a$) in the decision-making process of a hypothesis test.
  • The alternative hypothesis ($H_a$) plays a crucial role in the decision-making process of a hypothesis test. The goal of the hypothesis test is to determine whether there is sufficient evidence to reject the null hypothesis ($H_0$) in favor of the alternative hypothesis ($H_a$). The test statistic and the p-value are used to assess the strength of the evidence against the null hypothesis. If the p-value is less than the chosen significance level, the null hypothesis is rejected, and the alternative hypothesis is accepted. This means that the researcher has provided enough evidence to support the claim or statement represented by the alternative hypothesis. The alternative hypothesis, therefore, guides the interpretation of the results and the conclusions drawn from the hypothesis test.

Related terms

Null Hypothesis ($H_0$): The null hypothesis is the statement that the researcher is trying to disprove. It represents the assumption that there is no significant difference or relationship between the variables being studied.

Hypothesis Test: A hypothesis test is a statistical procedure used to determine whether the null hypothesis should be rejected in favor of the alternative hypothesis, based on the available sample data.

Test Statistic: The test statistic is a numerical value calculated from the sample data that is used to determine whether to reject or fail to reject the null hypothesis in a hypothesis test.